Hi, thanks for the question. It is a good one... but not always an easy one to answer. The reason for that is our enemy often tempts in areas that we are weak in. When the enemy comes against us it is usually exploiting the weakness of the flesh so it's not always just one or the other. Let me give you a couple of examples - one from the Old Testament and one from the New.
1. David numbers the Israelites
1Ch 21:1-3 Satan rose up against Israel and incited David to take a census of Israel. (2) So David said to Joab and the commanders of the troops, "Go and count the Israelites from Beersheba to Dan. Then report back to me so that I may know how many there are." (3) But Joab replied, "May the LORD multiply his troops a hundred times over. My lord the king, are they not all my lord's subjects? Why does my lord want to do this? Why should he bring guilt on Israel?"
David was incited or tempted by Satan to number Israel which even his not-so-godly commander Joab knew was wrong in the sight of the Lord and would bring guilt upon Israel. Yet David did it. You have an aspect of David's fleshly nature (in this case probably pride) being tempted by Satan. Now did David know that Satan was doing this? Not at all. But He was behind it, while using an aspect of David's fallen nature. If David had been right with the Lord at the time it wouldn't have happened. But Satan was subtly appealing to David's ego and fleshly confidence, leading to his sin.
2. Peter tries to correct Jesus!
Mat 16:21-23 From that time on Jesus began to explain to his disciples that he must go to Jerusalem and suffer many things at the hands of the elders, chief priests and teachers of the law, and that he must be killed and on the third day be raised to life. (22) Peter took him aside and began to rebuke him. "Never, Lord!" he said. "This shall never happen to you!" (23) Jesus turned and said to Peter, "Get behind me, Satan! You are a stumbling block to me; you do not have in mind the things of God, but the things of men."
Peter didn't want Jesus to die and sought to correct Jesus. The Bible says that Peter even began to 'rebuke' Jesus! Crazy thought! This came straight after Peter's great revelation that Jesus 'was the Christ, the Son of the living God' (Matt 16:15-16) And yet here we are straight after it and the fleshly concerns of Peter were somehow being used by Satan to try and thwart the plan and will of God. Hence, Jesus' rebuke saying 'Get behind me, Satan!' I'm sure this shocked Peter. Peter probably thought his concern sounded good. But he was being influenced by the enemy at this time. Would he have known it? Not at all. But like David above, Satan was using the an opening through Peter's flesh to exploit it. In this case probably part genuine concern for Jesus - though still advice against the Father's will, and part fear of what was coming - 'If Jesus was going to die, what would happen to his followers?')
So we see that there can be a cross-over between Satan and the flesh. He tempts and the flesh, in it's weakness, often picks up the temptation and runs with it. That is not to say that Satan or his henchmen are in on everything bad that we do. Some people blame everything on Satan and this is wrong. They don't recognise that our fallen nature is corrupt and a lot of the time it is simply our nature alone that leads us astray (James 1:14-15). Blaming everything on Satan leads some to try cast out so-called 'spirits' of 'greed, lust, pride, envy, jealousy etc' - Things that the Bible would put in the category of our fleshly fallen nature, not indwelling evil spirits,
What the New Testament says about how Satan attacks believers.
I went through the references (mainly New Testament ones) to 'Satan', 'Devil' etc to see again how Satan operates and they could be placed into four categories: 'Tempter', 'Deceiver', 'Accuser', 'Persecutor/Afflicter'.
Seen right from the Garden of Eden: Satan's temptations working on the desires of the heart (1 John 2:16)
Satan is seen as the tempter with Jesus: Matt 4:1 Then Jesus was led by the Spirit into the desert to be tempted by the devil.
Satan is seen as one who offers success in response to loyalty: Matt 4:8 Again, the devil took him to a very high mountain and showed him all the kingdoms of the world and their splendor.
1Cor 7:5 Do not deprive each other except by mutual consent and for a time, so that you may devote yourselves to prayer. Then come together again so that Satan will not tempt you because of your lack of self-control.
One who plants evil amongst the true: Matt 13:39 and the enemy who sows them is the devil. The harvest is the end of the age, and the harvesters are angels.
Can look like a minister of the Gospel: 2Cor 11:14 And no wonder, for Satan himself masquerades as an angel of light.
Able to deceive through miracles and signs: 2Th_2:9 The coming of the lawless one will be in accordance with the work of Satan displayed in all kinds of counterfeit miracles, signs and wonders,
Can speak through believers: Matt 16:23 Jesus turned and said to Peter, "Get behind me, Satan! You are a stumbling block to me; you do not have in mind the things of God, but the things of men."
Satan is the accuser - Rev 12:10 Then I heard a loud voice in heaven say: "... the accuser of our brothers, who accuses them before our God day and night, has been hurled down.'
Zec 3:1 Then he showed me Joshua the high priest standing before the angel of the LORD, and Satan standing at his right side to accuse him.
Tries to keep people in fear - Heb 2:14-15 '... He might destroy him who holds the power of death--that is, the devil and free those who all their lives were held in slavery by their fear of death.'
Can cause physical infirmities: Luk_13:16 Then should not this woman, a daughter of Abraham, whom Satan has kept bound for eighteen long years, be set free on the Sabbath day from what bound her?"
Can indwell unbelievers: Luke 22:3 Then Satan entered Judas, called Iscariot, one of the Twelve.
Desires to persecute believers: Luke 22:31 Simon, Simon, Satan has asked to sift you as wheat.
Can put roadblocks in the way of believers: 1Th 2:18 For we wanted to come to you-- certainly I, Paul, did, again and again--but Satan stopped us.
Seeks to destroy - 1Pe 5:8 Be self-controlled and alert. Your enemy the devil prowls around like a roaring lion looking for someone to devour.
John 10:10 The thief comes only to steal and kill and destroy; I have come that they may have life, and have it to the full.
What about the flesh? How to discern...
The Bible declares the acts of the flesh to be:
Gal 5:19-21 The acts of the sinful nature are obvious: sexual immorality, impurity and debauchery; (20) idolatry and witchcraft; hatred, discord, jealousy, fits of rage, selfish ambition, dissensions, factions (21) and envy; drunkenness, orgies, and the like. I warn you, as I did before, that those who live like this will not inherit the kingdom of God.
Mar 7:20-22 He went on: "What comes out of a man is what makes him 'unclean.' (21) For from within, out of men's hearts, come evil thoughts, sexual immorality, theft, murder, adultery, (22) greed, malice, deceit, lewdness, envy, slander, arrogance and folly.
So straight away we see a difference between Satan and the flesh. Satan attacks believers from the outside. The flesh comes out from within. Like I said above, there is some cross-over when Satan is tempting us in an area of weakness because that is the enemy and the flesh working together. But in other cases it is more obvious to differentiate (in generalities at least). Persecution is not listed as being the flesh - this is the work of the enemy through other people. Accusing and condemnation is not of your own flesh - likely the enemy operating through someone else or directly. Deception is not of your own flesh but is outward and from the enemy, working through people. Inward attitudes that plague us like pride, lust, greed, envy, coveting etc - these are the flesh. Again, Satan may tempt in this area but ultimately it is our flesh. If we are walking with God, walking in the Spirit and not in the flesh, then we won't have these problems.
How to respond
Flesh / Sinful nature
You are correct that Romans 7 and 8 is not about Satan but the flesh. And the beauty of the end of Romans 7 is that Paul comes to the end of himself and in doing so finds the secret! He wasn't trying to 'crucify the flesh' because he finally saw that he was the problem ('oh what a wretched man I am!') If he was the problem then he can't be the answer and the flesh trying to crucify the flesh simply won't work! Impossible! Thus Romans 8 starts with the great realization of two amazing truths - 1) There is no condemnation for those in Christ Jesus 2) There is a law of the Spirit of life that can set us free from the law of sin and death. That is, there is something that the Spirit of God can do in us and through us that can free us from this other law of always falling because of the weakness of the flesh. We are called to 'know' (Rom 6:6) that we were crucified with Him (as a side note the Bible never tells us to crucify the flesh - all references to our crucifixion with Him are past tense), and to 'count' or 'reckon' this to be true and see yourself as alive to God (Rom 6:11) and to present our body to God for His use and not to sin (Rom 6:13).
Rom 6:6 For we know that our old self was crucified with him so that the body of sin might be done away with, that we should no longer be slaves to sin.
Rom 6:11-14 In the same way, count yourselves dead to sin but alive to God in Christ Jesus. (12) Therefore do not let sin reign in your mortal body so that you obey its evil desires. (13) Do not offer the parts of your body to sin, as instruments of wickedness, but rather offer yourselves to God, as those who have been brought from death to life; and offer the parts of your body to him as instruments of righteousness. (14) For sin shall not be your master, because you are not under law, but under grace.
Rom 8:1-6 Therefore, there is now no condemnation for those who are in Christ Jesus, (2) because through Christ Jesus the law of the Spirit of life set me free from the law of sin and death. (3) For what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering. And so he condemned sin in sinful man, (4) in order that the righteous requirements of the law might be fully met in us, who do not live according to the sinful nature but according to the Spirit. (5) Those who live according to the sinful nature have their minds set on what that nature desires; but those who live in accordance with the Spirit have their minds set on what the Spirit desires. (6) The mind of sinful man is death, but the mind controlled by the Spirit is life and peace;
Gal 5:16-17 But I say, walk by the Spirit, and you will not carry out the desire of the flesh. (17) For the flesh sets its desire against the Spirit, and the Spirit against the flesh; for these are in opposition to one another, so that you may not do the things that you please.
In terms of Satan's attacks, we are called to resist as you say but how we do this depends on what approach he is taking. Deception (or coming like a serpent), is different from open persecution (or coming like a dragon). Temptation is different again. Some key passages would be:
Mat 4:3-4 The tempter came to him and said, "If you are the Son of God, tell these stones to become bread." Jesus answered, "It is written: 'Man does not live on bread alone, but on every word that comes from the mouth of God.' "
2Co 2:11 If you forgive anyone, I also forgive him. And what I have forgiven--if there was anything to forgive--I have forgiven in the sight of Christ for your sake, in order that Satan might not outwit us. For we are not unaware of his schemes.
Eph 4:26-27 In your anger do not sin : Do not let the sun go down while you are still angry, and do not give the devil a foothold.
Eph 6:11 Put on the full armor of God so that you can take your stand against the devil's schemes.
Jas 4:7 Submit yourselves, then, to God. Resist the devil, and he will flee from you.
1Pe 5:8 Be self-controlled and alert. Your enemy the devil prowls around like a roaring lion looking for someone to devour.
Obviously this is a big topic but hopefully there is some points there that help and stand out.
May God Bless!