What I see here reminds me of an article titled The Emotional Terrorist. He is using promises "to curb his temper and alcohol problem, and be a better man, husband, and father...to try to work things out with me and save our relationship" to manipulate B.A.AM into staying when she's already decided she needs to leave. He is the adulter, the drunk, the verbal abuser. Yet he feels entitled to dictate "numerous conditions" for continuing the relationship? It seems to me that things are already getting worse. B.A.AM has already considerably compromised her beliefs for him, now he demands she do so even more (when he's the one who should be groveling: what nerve!).
There is nothing to gain by negotiating with such people, there is no middle ground: whatever you give they take, offering little in return but empty promises and demands for further concessions.
I don’t know if I can do this. I don’t think I have the faith in his promises that this for the better of our family
That doubt is the voice of reason saying what the emotions refuse to hear!
B.A.AM, if you're reading here: see a lawyer about getting child support and custody orders. You need to look out for the welfare of the children, even if there is no marriage to dissolve.