Romans 7:20 - Are we responsible for sin?

Romans 7:19-20
For what I do is not the good I want to do; no, the evil I do not want to do--this I keep on doing. Now if I do what I do not want to do, it is no longer I who do it, but it is sin living in me that does it.

From this passage, it seems that Paul is saying that he is not responsible for his sin. As if, since he doesn't want to do it, it isn't actually him who does it. Of course, this can't be the explanation, since we are responsible for our sin and will be held accountable for it (Ezekiel 18:20, Revelation 20:12-13, Ecclesiastes 12:13-14 and others). The bible is God's word (2 Timothy 3:16-17) and therefore cannot have errors or be wrong, so there must be another explanation. Does anyone have any thoughts/suggestions? Feel free to comment!

In thinking about it within the flow of the book of Romans, heres an explanation that may fit. I don't know if it's right, or if it even makes sense, but it's an idea.

 Gill says that this is a matter of the new nature we have in Christ (Romans 6) verses our old sinful nature. Once saved, we are a new creation in christ (see 2 corinthians 5:17). As a new creation, we are no longer stuck in sin, but forgiven - we are no longer under wrath but under God's grace. Therefore, if we sin and our new nature is responsible then it's the same as the old - sinful and depraved. 1 John 3:9 says "No one who is born of God will continue to sin, because God's seed remains in him; he cannot go on sinning, because he has been born of God."  and yet we see that we still struggle with sin as Christians and fail so often. How do we explain this? Could it be that when we sin as Christians, our old nature is accountable, and Christ paid for that sin when he died and we were saved? It's a bit of a hard concept and I may be completely wrong, but When we sin, is our sin the fault of our pre-conversion self, and therefore has been dealt with by Christ already, resulting in our new nature not being responsible? 

Either way there is complete forgiveness in Christ - from our sin before we were saved, and from our sin after we were saved - he takes it all so that we are forgiven. I'm in no way arguing that we cannot be Christian if we struggle with sin after we are saved - thats the point of sanctification. But in Christ there is complete forgiveness, we are Justified (Just as if I'd never sinned).