I am probably going to be skewered for saying this - but in every single post you make about your husband, you are on the fence. Questioning his every move, word, motive, intention behind action, etc. Ok, so you've established you don't trust him. Yet you don't want to take the steps to end this relationship. Here's my suggestion: get off the fence and commit. You have two choices here: stay or go.
If you stay, of course he is going to want to have sex. It is how men show their love - it is an action. Yet you treat this as if its some kind of insult that your husband wants you in this way. Read some Dr. Laura books on marriage - withholding sex is a marriage killer. If you stay, accept him for who he is. Doesn't matter if he's an alcoholic, or bi-polar, or both. Doesn't matter why. It is what it is. He is who he is. Sure, everyone has the potential to "change" to become what we think we want them to be. But will they? Highly unlikely. So find the acceptance in what it is today,and wither be ok with that - hell find the good in it all - or leave.
Find some forgiveness. He's a human being after all. We all make mistakes and royally f-up sometimes. If you can't find forgiveness, leave.
Treat him the way you want to be treated, instead of remaining withdrawn as some form of manipulation to get him to "change". Maybe if you took the initiative here, he'd follow. Maybe you hold way more power over the tone of your marriage than you really think. If this doesn't appeal to you, leave.
Being halfway into anything is a recipe for failure. How could it not be? You're not committed when you're only halfway into something. So make a decision and commit to it. It will either work out or it won't. But you'll never know that until you try it.