Welcome to SR 3!
I also agree with Suki and MissFixit. I was married to a cheating, lying AH. My next relationship was with my now exADFH (ex alcoholic defacto). He couldn't stand cheating. He was always very loyal and I valued that in him so much. The first A would blame me for everything and I never knew where he went when he did his 'disappearing act' or who he was with. In his current relationship, he recently, took off for a week, screwed someone I know, came back and lied to his partner saying that he went to a club, and someone must have drugged him because he can't remember much about what happened next except that he may have had sex with someone, but can't really remember if he did or not due to the tranquilisers that were dropped in his drink!...he's such a compulsive liar.
Hypothetically, if the second one had ended up having sex with someone in a drunken state, I know he would have admitted it, wouldn't have blamed the alcohol, would have felt terrible about it and probably wouldn't have forgiven himself. He would never have had an affair. All in all, two alcoholics with two different personalities and two differing sets of values.
As they say in AA, alcoholism is a three-fold disease. It's physical, emotional and spiritual. If your AH only removes the alcohol and doesn't address the emotional and spiritual part of the dis-ease, his character defects will remain and this will show up in his behaviour and actions. While you're giving yourself the time to process the situation and feelings, keep sharing and reading here. There's a lot of support and experience on these boards. I know it's helped me. You may also find Al-Anon meetings helpful too. All the best 3.