I believed my AstbXH when he said he would not drink during his Wed night and EOW visitation with the kids (7 & 9). I suspected he would drink after they went to bed (which is still unacceptable to me because DD has a seizure disorder and has many at night - passed out drunk dad is not OK and could be deadly) He denied it. He was drunk for EVERY evening phone call to the kids when I had them. Eventually he went to work on a weekday drunk, drank more at lunch, then got fired. Now....if he can't go to work (very corporate environment with a lot of meetings and contact) where other adults are likely to notice, sober, is he really staying sober in front of his 7 & 9 year old kids who don't have a clue about alcohol use and its effects? No friggin way. The longer he was in an environment where I was no longer "the booze police", nagging wife, enabler, his alcoholism progressed rapidly. He became a danger to my kids, himself, and everyone around him.
You cannot believe him when he says he will not drink during the days he is with your son. He WILL lie, especially since he does not acknowledge that he has a problem.
I know that the thought of being a truly single parent 24/7 is overwhelming, but at this time, it just does not seem like a safe situation for your son.