A previous verse in the Psalm makes me think we need to look long and hard about what David is saying.
"Against you, and you only, have I sinned"..
Are we to take this as saying we can only sin against God, and not against our fellow man? If we "go in to" another man's wife [KJV innuendo again!] are we not sinning against that man? Does the Lord's prayer not ask us to "forgive those who sin against us"?
I think a look into the original language could be useful. What exactly does the word "sinful" in Hebrew mean? The KJV also uses debts or trespasses to talk about sin. Do these originate from different Hebrew words?
Could it be in this context more about a sinful nature, of being capable of sin, rather than actually having sinned before birth?
Just to feed the speculation.........
God Bless
Nick