I haven't been through a child custody battle with an alcoholic (or anyone) so I'm just going to give you my opinion here.
You are hoping to sit, quietly, on the fence until he gets his money and you can get him out of there. There is no reason, based on what you have said, that you can't still do that.
I get that you don't want to rock the boat.
First off, take a deep breath, I think you will find that visitation for an alcoholic isn't as pressing as it seems once they are off on their own. Right now it is at least a bargaining tool for him (I don't know what he's like so I'm guessing).
If he is in fact an engaged Father (as much as an alcoholic can be), then he may truly care.
Regardless, once he is free to drink at will, visitation will not seem that attractive probably (sadly).
But all of this can wait a few days correct? Until you get those financial papers signed, as soon as possible.
In the meantime, no, he shouldn't be driving your child anywhere, is there any way you can manage that? He should not be with your Son unsupervised. If he is looking after him at home (is this avoidable?), I would call every hour and speak to your Son. But how do you keep him out of the car. You may well have to have a talk with your stbxAH about it, I personally can't think of any other way, can you? Short of sending your Son away for a week to relatives for a holiday or something, until the financials are done - I don't know what options you might have?
Once the financial agreement is signed and sealed, well before even, you can perhaps speak to your lawyer and get their legal opinion and help, that is really the only way to proceed. Perhaps soberlink would be a viable option.