Welcome to SR. So sorry your going through this though the story is not new it still hurts all the same.
My thought is if he wants to be part of the family he has a lot of work to do first, since he is already moved out there is no reason to put yourself or your children through any more. Maybe look at it like you have just met this man, because that is what is really happening. They say in the rooms of recovery not go get involved in a relationship for a min of a year when they get clean and sober. So give him his year to get help and you can keep your focus on yourself and children.
His recovery is not your job to help and to be honest if he is not doing the work to find help and get sober for himself he is not ready and will only drag you deeper. Don't try so hard to help him, instead just let him help himself and maybe you could seek out an Alnon meeting and get support for yourself
Looking forward to getting to know you better, stick around and keep reading