I think that if he has been backing out of meeting with you for two weeks, you have your answer about his priorities at this time. And as I write that, it sounds so harsh. But truth often hurts. I am an alcoholic and have been sober for a couple years now. My husband is also an alcoholic, currently sort of maybe trying to stay sober. He was back at it last night after having almost two months sober time.
Your partner needs to want this for himself. It doesn't appear that he does right now. My husband and I also have a son. When our son was two my husband was in a drunken rage and insisting that our son wasn't his child. Perhaps that is something he needed to tell himself to justify his anger and continued drinking. I don't know.
I'd suggest what Jeni said. Family and Friends of Alcoholics Forum here on SR. AlAnon meetings. You need support. And take this time apart to decide what to do. Maybe things could get better but do you really want to wait for that to happen? Life is too short. My son is ten now and the husband is still struggling back and forth with recovery.
Take care of you. Take care of your child.