What is Paul saying in Romans 7:15 - 7:25? The way I interpret it is that he's saying that if we commit sin while we are aware that it is sinful and if we hate the sin we commit then we are not really committing sin but rather the "sin which dwells in [us]" is doing it. Is this the generally accepted interpretation?

The NASB text:

14 For we know that the Law is spiritual, but I am of flesh, sold [m]into bondage to sin. 15 For what I am doing, I do not understand; for I am not practicing what I would like to do, but I am doing the very thing I hate. 16 But if I do the very thing I do not want to do, I agree with the Law, confessing that the Law is good. 17 So now, no longer am I the one doing it, but sin which dwells in me. 18 For I know that nothing good dwells in me, that is, in my flesh; for the willing is present in me, but the doing of the good is not. 19 For the good that I want, I do not do, but I practice the very evil that I do not want. 20 But if I am doing the very thing I do not want, I am no longer the one doing it, but sin which dwells in me.

21 I find then the [n]principle that evil is present in me, the one who wants to do good. 22 For I joyfully concur with the law of God [o]in the inner man, 23 but I see a different law in [p]the members of my body, waging war against the law of my mind and making me a prisoner [q]of the law of sin which is in my members. 24 Wretched man that I am! Who will set me free from [r]the body of this death? 25 Thanks be to God through Jesus Christ our Lord! So then, on the one hand I myself with my mind am serving the law of God, but on the other, with my flesh the law of sin.
Paul brings up covetousness as a personal example of the tension he is about to explain, which tension is brought about through the new birth.  "When the commandment came", Paul says, "sin came alive and I died" (Romans 7:9).  Now we know that Paul was raised not only as a Jew but also as a Pharisee (Philippians 3:5-6) so it is clear that he is not referring to the first time he was exposed to the 10th commandment; he had known the words his entire life.  

When the new birth ushers one into the Kindgom, the commandments come in a new way.  The Spirit of Christ, the living Word, living in and renewing minds through obedience brings the commandments to a newly cleansed conscience and sin is revealed for what it truly is: "Did that which is good, then, bring death to me? By no means! It was sin, producing death in me through what is good, in order that sin might be shown to be sin, and through the commandment might become sinful beyond measure." - Romans 7:13.

Sin is in view in the rest of the passage, not so much as specific instances, but as a governing principle lodged in his humanity.  It is this principle that came to power when the commandment came and he died; not because he suddenly coveted but because the newly illuminated Law revealed how covetous he already was.  The Law is the opposing principle and though it is good and holy it is too weakened by sinful flesh to bring about righteousness: "For we know that the law is spiritual, but I am of the flesh, sold under sin.- Romans 7:14"  Paul could not rid himself of the covetousness
he now abhorred.

Paul has a desire in his renewed mind to do that which is good but does not find within his natural self the ability to carry those things out.  Instead he finds himself doing what he does not wish to do (Romans 7:15).  All sin is birthed at the level of intention which is made clear in the Sermon on the Mount.  He cannot attribute this failing to the presence of the Law (which shines the light upon sin but cannot alleviate it) and so he attributes it to "sin that lives in him." - Romans 7:16-17  He is not dodging culpability but outlining his moral failure at the level of intent.  His responsibility is to consider himself dead to sin but alive to God; essentially to believe the Gospel and act accordingly, relying on the mercy and grace of God every step of the way.

The closing statements reveal where his hope lies in the midst of this tension.  It is not in his agreement with the Law nor in his inner (and newly spiritual) desire to do that which is good.  It is not in his natural ability since, in that arena, he is sold under sin.  "For I delight in the law of God, in my inner being, but I see in my members another law waging war against the law of my mind and making me captive to the law of sin that dwells in my members." - Romans 7:22-23  His hope lies in Christ who both satisfied the Law on his behalf through his obedience and condemned sin in the flesh through the cross:  "Wretched man that I am! Who will deliver me from this body of death? Thanks be to God through Jesus Christ our Lord! So then, I myself serve the law of God with my mind, but with my flesh I serve the law of sin." - Romans 7:24-25

Paul is able to live in this tension where the flesh and Spirit lust against one another free of condemnation by faith in the work of Christ and the sanctifying presence of the Spirit.

"There is therefore now no condemnation for those who are in Christ Jesus. For the law of the Spirit of life has set you free in Christ Jesus from the law of sin and death. 
For God has done what the law, weakened by the flesh, could not do. By sending his own Son in the likeness of sinful flesh and for sin, he condemned sin in the flesh, 
in order that the righteous requirement of the law might be fulfilled in us, who walk not according to the flesh but according to the Spirit." - Romans 8:1-3

His trust is: "And I am sure of this, that he who began a good work in you will bring it to completion at the day of Jesus Christ." - Philippians 1:6