Paul only used the term "spirit" once in Ro 7 (Ro 7:6), a chapter that described the hopelessness of people to ever keep the righteousness of the Law in their own strength. In contrast, the word "spirit" (or "Spirit") is used twenty-one times in Ro 8, a chapter that gives the answer to the hopelessness of Ro 7.
In these verses of Ro 7, Paul was not describing warfare that wages between the new man and the old man. He was contrasting the complete inability of people to save themselves because of their corrupted flesh versus the life-transforming power of Christ described in Ro 8.
The Apostle Paul was not living a life of constant failure where the good that he wanted to do, he was unable to accomplish, but the evil that he didn't want to do, he did. He wasn't living that kind of life because it was no longer him living, but Christ living in him (Ga 2:20). Christ in Paul was manifesting holiness in Paul's life that was second to none.
However, if Paul had abandoned his dependency upon Christ and had started trying to live the Christian life out of his own resources, then the condition described in Ro 7:15-24 would have been his experience.
Our flesh has been corrupted through sin, and though we can renew our minds through God's Word (Ro 12:2), we can never elevate our flesh to a place where it can fulfill the Law of God. Hence, the good news of Ro 8 that what the Law couldn't do, because of the weakness of our flesh (Ro 8:3), God did for us, and all we have to do is receive by faith.
"I know that in me (that is, in my flesh,) dwells no good thing" (Romans 7:18)
When Paul used this parenthetical phrase, "that is, in my flesh," he was specifying the natural (physical) part of his person. He was stating that in himself, apart from his born-again spirit, there was no good thing. He had to include this explanation, or his statement would not have been accurate, because in his spirit there was a good thing (Christ). Many times scripture speaks of "Christ in us." Paul clearly separates the flesh & the spirit in this scripture.
Paul says that "with the flesh he serves THE LAW OF SIN" in Romans 7:25 but in Romans 8:2 he says, "Christ Jesus MADE US FREE (past tense) FROM THE LAW OF SIN." To summarize all this, We are free to walk by the spirit but if we try to walk by the law we will be walking by the flesh. You can choose to serve the law of sin with your flesh (offer your members to the flesh) or you can choose to serve the law of the Spirit of life in Christ Jesus (offer your members to righteousness & put to death the deeds of the flesh).
Our old self WAS (past tense) crucified with Him (Ro 6:6)
The old self is dead and gone, but it left behind a body with attitudes and emotions that still influence us until we renew our minds. We are not dealing directly with the "old sin nature" but with its influence that is still being exerted through our unregenerate flesh. So the Christian life is a renewal of our minds to who we HAVE BECOME in Christ, not a hatred for who we are in our "old self."
Who shall deliver me from the body of this death? (Romans 7:24)
Paul didn't say, "Who shall deliver me from this death?" for the Christian has already been delivered from the death that is the wages of sin. He made special mention of the body of this death. The terminology "the body of this death" corresponds to what Paul called "the body of sin" in Ro 6:6, "Our old self was crucified with Him, in order that our BODY of sin might be done away with". He was not speaking of the sin nature itself, for a Christian no longer has a sin nature, but he was rather speaking of the lingering influence of the sin nature that still exerts itself through the un-renewed mind. So death, or the "old man/self," is gone, but the body that it left behind (i.e., the thoughts, attitudes, and emotions) still poses a problem to us as Christians. How do we overcome this flesh? The answer is stated in Ro 7:25 (God through Jesus Christ our Lord) and then explained in Ro 8.
Taken from http://www.awmi.net/bible/rom_07. Some words are my own.