IIn fact, I believe addressing such sins as addiction is far to simplistic. Addressing it as sin includes facing addictive behaviors, but it goes beyond this surface approach and to the root. In fact, I do not know of any passage of Scripture that addresses drunkenness and such as “addictions” rather than sins. Do you?
Galatians 5:16-21 comes to mind. Addiction is an issue of formed character. That's not to say that persons who have certain weaknesses are unredeemed, but they have not yet be properly formed in Christ.
Paul contrasts the character/lifestyle of a person captive to the flesh (the most severe form is addiction) and the person led by the Spirit. Regarding those who have an unredeemed character:
Galatians 5:16-21
Live by the Spirit, I say, and do not gratify the desires of the flesh. For what the flesh desires is opposed to the Spirit, and what the Spirit desires is opposed to the flesh; for these are opposed to each other, to prevent you from doing what you want. But if you are led by the Spirit, you are not subject to the law. Now the works of the flesh are obvious: fornication, impurity, licentiousness, idolatry, sorcery, enmities, strife, jealousy, anger, quarrels, dissensions, factions, envy, drunkenness, carousing, and things like these. I am warning you, as I warned you before: those who do such things will not inherit the kingdom of God.