Either Al-anon or therapy is going to be useful for you.
Why would you stay with someone who is an alcoholic, who you don't trust and who cheats on you (and yes you know his lies are all B.S.)
You could have really low self esteem and feel like you don't deserve any better or you might be afraid of being alone. But right now you're acting like a doormat and he is walking all over you, sweetie. You will find someone better, than this, trust me.
Now just kick him to the curb!
