A couple of possibilities:
One is supervised access, where he spends time with the kids but there's another (sober) adult present all the time. The supervisor could be yourself (fun!) or someone else that you both agree on. Some places offer supervised visitation centres, where parents in your situation bring kids for time with a parent who is not able to look after them on his/her own. Ask your attorney about this.
Another option is visitation in a public place (library, park, etc). You bring the kids there and he hangs out with them for a hour or two (probably not much longer, given their ages). If he's drunk, you can leave with the kids.
Given their age and the fact that they're too young to alert an adult if things go badly, I think it is reasonable to say that you don't want them staying at his place until you have reason to be confident that ex will be sober and responsible the whole time (completion of addiction treatment or parenting courses or something - yes, I know these aren't bulletproof but at least it's better than the situation you have now).
Don't be guilted into "the kids need their dad" (which has happened to me). The kids need a responsible and responsive parent, not some drunk snoring on a couch. Until you can be reasonably confident that they will get the former and not the latter, I think you are in good shape to tell your lawyer you don't want them alone with him.