Hello and Welcome,
I am sorry that you are going through this pain. You sound like you have a good head on your shoulders. I will not lie, staying in a marriage after infidelity is extremely difficult and pianful and when other addictions are involved it is even harder.
You have done the right thing by seeking counselling. It doesnt always help though, so be warned. Things will get better but the WS must look deep inside of themselves and figure out why they cheat and why (in your H's case) they drink. He obviously has some issues of his own that has nothing to do with the marriage itself and do not take the blame for that. Many WS's say they are remorseful and beg for a second chance, but the fact that he has a history of cheating and alcoholism is not good, and sounds like this pattern will repeat if he does not do the work necessary to resolve his own issues. Many times it is best that each of you go to individual counselling before marriage counselling or in conjuction with the other.
Come back often and ask questions or just vent. And let us know how the counselling goes. We are all here to support each other. Take care & hugs.............Carol~