If I was to flip to the other side and say I once was the son of an alcoholic dad to give you some reassurance.
All children I think at some stage in life realise their parents aren't going to be around forever, and rather than never talking to a parent again and waiting for that awful thought of receiving a phone call one day, they mellow and make contact as there is still a connection through blood, I'm sure there are some exceptions, but if you're sober and your son sees you have changed that will be even more of a reason for him to make contact.
It's gonna take time though, not sure how old your son is, but he needs to figure out who he is, by that I mean finding his own identity in this world, something that only comes with maturity, especially now with starting his own family, until he does that, he won't realise how important family is in the bigger picture of life.