Just because he says he wants to get better doesn't mean he's willing to do the work to get there. If you read more postings here, A's often say what we want to hear.
It can be heartbreaking to go through cycle after cycle of this type of hope ("he really wants to change") followed by more drinking.
I agree--at 4 months I would question why you'd expose yourself to this. He needs to do this, and living together now is probably not in your best interests and won't make or break it if he wants sobriety.
And no, there really isn't anything for you to do here. Thinking you can change things is an illusion that will keep you stuck.