If I have an addiction, can I still be saved?
Yes! We are not called to perfect ourselves before we come to God. If we could do that, then Jesus could have spared himself the agony of dying of the cross in payment for our sins. He would have simply instructed us to live sin-free lives. He knew, though, that man is utterly incapable of cleansing himself, as demonstrated throughout the Old Testament.
However, once a person becomes a believer, the Holy Spirit begins to work in his life to convict him of those things that are unpleasing to God. The natural response to the saving work of Christ is to rid one's life of anything that brings displeasure to the Father.
This is a process, sometimes a long one. Some denominations claim that if you don't clean your life of sin completely, upon salvation, then your salvation wasn't real. This is simply not true. (See the question, “If I sin, does that mean I was never saved?”) Addictions are tricky, though. God wants us to depend on Him, not on some outside substance. Is it vital to your growth as a believer that you gain control over these life-controlling issues. You may need outside help for this. That is all right! Seek help through a trusted pastor, Christian counselor, or believing friend who will hold you accountable.
Satan uses addictions to show us that we are unworthy and that God can't help us. Don't listen to those lies. Recognize that your new identity in Christ is “Child of God.” Focus on building that identity and set aside the old one—the alcoholic, the gambler, the liar, or whatever your addiction may be. You may continue to struggle; that just may be the way that God wants to work in your life on a continual basis. Just keep turning it over to Him.
Philippians 1:6 “[B]eing confidant of this, he who began a good work in you will carry it on to completion until the day of Christ Jesus.”