welcome to sr, uncertainty.
reading between the lines here, but it sounds a little like you are entertaining a notion that you may reconcile with your husband. my word of caution would be: don't put the cart before the horse.
the addict in my life lied to me after getting sober. he is just emotionally very immature (not name-calling; he really is) and when that is going on, he cannot function like a mature adult would who believes that "I did a bad thing". Instead he believes "I am bad" like a 5 to 10 year old would. So the lying thing would kick in when there was something shameful involved. He continues to get more honest, but it's a slow road.
the more sobriety, the more the self-esteem increases, the more he accepts that he is a flawed but still worthy human being, and the more he truly wishes to have healthy relationships, the more it should improve. in the meantime, i have moved on.
IF your husband is truly remorseful, and wishes to make ammends, more power to him. You have no reason nor obligation to resume your relationship, move in together, or even start having conversations. if i were in your shoes, i would meet with him, as it could be a healing thing to hear what he has to say. but whatever you think, hope, wonder, about your future together, proceed very slowly