John 8:34-36, "Jesus answered them, Verily, verily, I say unto you, Whosoever committeth sin is the servant of sin. And the servant abideth not in the house for ever: but the Son abideth ever. If the Son therefore shall make you frees, ye hall be free indeed."
It is true that we are born with a sin nature, but when you receive the free gift of salvation through Christ alone, He MAKES you free. Sinning, then, is not a result of addiction, but rather, you are held in addiction as long as you choose to sin.
The solution is very simple for the Christian: yield to the leadership of the Holy Spirit in your life, and don't sin. Galatians 5:16 says, "This I say then, Walk in the Spirit, and ye shall not fulfil the lust of the flesh."