So it would seem, after working through Romans 6-8, that Paul does not in any way teach that sin is a normal part of a believer's life, despite justification by faith. Indeed, this is exactly the accusation he has been trying to fend off! Rumors are circulating about him that he teaches, "Let us do evil that good may result" (Rom. 3:8). People are saying he teaches sin is a good thing. He has worked hard in Romans 6-8 to show that this accusation is false. His message is that we cannot continue sinning (6:2, 15), that those who are "in the flesh" cannot please God (8:8).
But after we have finally heard Paul and shaken off the popular misinterpretation of Romans 7, we are still left with the problem of application. The reason Christians have been able to ignore Paul's repeated statements on sin in Romans 6-8 is because they identify so much with the second half of Romans 7: "what I want to do I do not do, but what I hate I do" (7:15). Experience trumps the surrounding two chapters. How then do we deal with the pervasive sense of sin so many have in their lives?
In this regard, the Wesleyan tradition from which I come has always been more optimistic than other traditions, at least on the books. At least in the past, we have preached that God can perform miracles in your spiritual life just as we believe he does with physical sickness. But "holiness," as our tradition used to call this hope, could also degenerate into petty legalism over what a person wore or how they did their hair. At one point in the mid-twentieth century, many of those who most tauted holiness ironically were thoroughly sinful in their attitude toward others and did almost no good at all in the world.
Part of our pessimism is how introspective our world has become. We are probably more aware of our inner feelings than any generation in any time or place prior to the 1800s. We experience intensely as Jesus reveals that the heart is the source of our actions or when Paul prioritizes faith over works. But we probably experience these shifts more intensely than Jesus and Paul themselves spoke them, and we can hyper-analyze our motives well beyond anything they intended. Our standard of living above sin goes far beyond the biblical expectations. We think anything short of absolute perfection is sin. The New Testament standard is much closer to John Wesley's sense of sin as intentionally doing wrong. 
Then there are addictions and tendencies. Our strengths often have corresponding weaknesses. If we are a decisive and assertive leader, we can have a tendency to run over people. If we are compassionate and understanding, we can have a tendency not to help others develop discipline or empower their weaknesses. Sometimes we develop addictions we could never overcome in our own power. We have developed a keen sense of these sorts of things in our last two centuries of introspection. While the New Testament understands slavery to sin, its world did not think in terms of genetics and environment.
Paul's argument against sin in Romans 6-8 was a defensive one. He was making it clear that his theology did not encourage or excuse sin. Because he taught a person could not get right with God by keeping the Law (focusing on the Jew-specific parts), he needed to make it clear he was not excusing violation of what we often call the "moral" law.  This debate is far removed from the issue we are discussing now, to what extent sin is a normal part of a Christian's life.
1 John has the mix just right. On the one hand, anyone who thinks they have never sinned does not understand the human condition (1 John 1:8, 10). Yet intentionally doing wrong cannot be normal for a Christian. After all, we have God's "seed" inside us (1 John 3:9). Nevertheless, Christians do sometimes sin and when this happens, we need to seek God's forgiveness through Jesus Christ (2:1)...
 One can, of course, wrong others without intending to do so, and one can have an evil heart without realizing it, as has sometimes been the case of those who most preached victory over sin.
 The biggest problem with understanding Paul here is that he uses the word "Law" to refer to both and what we call the "moral law" was not a category Jews used--it is a later Christian way of processing what parts of the Old Testament Law we still keep and the parts we do not.