Wow your husband sounds so much like my partner did a while back, and it really peeved me off back then. In my case I think it was like you, a case on my part of having a history of chronic relapse and lying over alcohol. I am typically a very honest person, have never cheated, blah blah blah. But ask me about whether I was drinking, how much or when and I would lie through my teeth. That really takes a toll on a relationship!
There is really only one way we can change our loved ones minds. Stay sober. Be the honest, kind people they fell in love with- only more grown up. Keep it up, be patient. It may take months or more than a year, depending on him and how deep the hurt is. But he will come around. Things will keep getting better along the way.