Most of us, when we think of sin, think of "little s" sin. "Little s" sin is a moral failure, a "missing the mark."
To be sure, Paul talks about "little s" sin and considers it a problem. But according to Paul, our real problem with sin isn't with "little s" sin but with "Capital S" Sin.
There's a difference between sin and Sin.
"Capital S" Sin is a force that enslaves human beings. According to Paul, "Capital S" Sin is a power we submit to and serve, like a god.
For example, Paul says in Romans 6:
Let not sin therefore reign in your mortal body, to make you obey its passions. Do not present your members to sin as instruments for unrighteousness, but present yourselves to God as those who have been brought from death to life, and your members to God as instruments for righteousness. For sin will have no dominion over you, since you are not under law but under grace.This is a description of "Capital S" Sin. Sin isn't a mistake. Sin is a power that can reign and rule in your body, forcing you to obey. Sin can have dominion over you. Sin is a false god to whom we can give idolatrous allegiance. We can present our bodies either to God or to Sin.
All that to say, if you don't understand Sin as an enslaving power you'll never understand Paul's vision of salvation.
Salvation isn't merely about being forgiven.
Salvation is about being set free from the power of Sin.