Dina,
I agree with Susan that if your husband believes he is an alcoholic, that is all that matters and you should do what you can to support him. I think it's strange that he tells you he has other problems besides alcohol that he can't talk to you about. I would encourage you to tell him it needs to come out- in counseling or wherever. It could be that he is so focused on dealing with his alcohol issue (which is a good thing) that he is avoiding other issues that affect the relationship. He needs to focus on both (and so do you).
I haven't attended AA but I've heard that they encourage making amends to those who you have hurt but not telling about affairs because the theory is that this would cause more hurt to the other person. I don't agree with this at all (if I understand it correctly... maybe others can tell me if I'm misinterpreting it or misquoting it), I'm of the belief that an affair is hurtful even if the other spouse doesn't know about it and hangs over the marriage and that it needs to be addressed for the person to recover and the marriage to recover. (Granted I'm not married, just going on my own personal opinion of marriage/relationships and experience with my parents' marriage etc. It is just my belief. I would want to know and also believe the truth has a way of coming out and I would be hurt more by not knowing and God forbid finding out some other way than by my spouse confessing to me and asking for forgiveness). I'm not saying your husband had an affair and I definitely think he should tell you if you asked him and he said no. But maybe he thinks telling you would hurt you more. I'm totally just speculating here but the important thing is getting him to be honest about whatever these other problems are... I really don't think the marriage can improve, or he can improve really, without addressing those.
Maybe they will have better advice in the marriage and family section, I don't know. I really don't get not agreeing that someone who thinks they are an alcoholic is not an alcoholic. To me it seems like a very personal thing and who is anyone to question anyone else about it, even a spouse. I'm sure some people have suspicions that I'm an alcoholic or a problem drinker-- the people who drink with me!-- but many other people would have no idea. I have hid how much I drink from people closest to me so as not to be told to stop. And I also think there are different levels of alcoholism- people are affected differently by different quantities and types of alcohol etc. To me if someone thinks they are an alcoholic, they are and they need to address it and I don't feel it's right of you to question your husband's concerns about himself like that. I don't see how AA can ruin a marriage-- I see how it can look religious-y or cult-ish (I'm sure it's not but that's the picture I have of it for some reason and I can see how others/you might see it that way as well), and I don't agree with the position on not revealing affairs and that is really the only thing I could see it directly affecting you and your marriage- otherwise if it helps him and does good things for him then how does it affect your marriage negatively, I don't understand. But I wish you luck.
It's kind of hard to answer your questions because I am struggling with dealing with my own demons and I need support from my family members and friends- if anyone close to me questioned my need to recover or my belief that I have a problem drinking, I would be very hurt. I need their belief in me and their unwavering support. In fact I am too cowardly to tell many people as of yet so if they didn't take me seriously or tried to dissuade my efforts I would be crushed. :-( Maybe that right there is what is affecting your marriage so badly. I feel you should give your husband the benefit of the doubt and really believe with him that he has this problem and that he is working on fixing it- help him instead of detracting from his goal, and maybe things will get a lot better. Good luck.