Sounds like you've hit your rock bottom. People around here and at al-anon teach the 3Cs. You didn't cause his drinking, you can't control his drinking and you can't cure it. You can read around the forums here gain some sense as to why your alcoholic BF thinks you are cheating, is constantly lying (living a lie in fact) and more.
What you can control is you. When we are in a relationship with an alcoholic we also get sick, and weak. You aren't doing him or you any favors by staying.
People also speak about detaching here, being able to walk away from the mess. The point of this is to become stronger and start taking care of yourself again.
You must be pretty convinced you are in a bad situation that isn't getting better. No more talking, no more arguing, no more engaging him, no more stressing about it, it's decision time. Do I put up with this hell hole or do I exit.
How do you let go? By focusing on you and not the relationship. Take time to read all the posts here, they are motivating and will help get you on the right track.