FWIW I do think your ex sounds abusive during your marriage and has possibly contributed towards your alcoholism escalating at that time.
I think he is partly responsible for engineering the situation where you couldn't keep the DC, possibly to punish you for leaving him. Certainly reducing his earnings to avoid paying much child maintenance at that time wasn't him being a good dad or a reasonable person at all. You also contributed to that situation, firstly by being alcoholic which will have made you struggle to be a good parent, and meaning secondly that money you needed for life's essentials went on alcohol, contributing towards you not being able to afford to keep the DC.
I don't see you as typically "a cheater". It sounds like you had an exit affair due to feeling controlled and unable to therefore simply leave. As if you felt you had to create a reason for him to not want you?
And I think your ex is still trying to exert his control now.
Unfortunately for you, I don't think entering into a legal battle with an abusive person whilst you are trying to recover from childhood trauma is going to be in your best interests for recovery. It could push you back into alcoholism through stress.
In the interests of balance I will say that I don't believe there is such a thing as an alcoholic who isn't abusive in some way (often emotionally. And emotional abuse can be equally or more so as damaging as physical abuse). I think the abusive traits come as part of being an alcoholic. What I can't say is whether abusive alcoholics were abusive before they became alcoholic, because I've only ever met these people at the alcoholic stage so I haven't known them sober. So I'm not calling you an abusive person in general, I don't have the information to judge that.
I do think you're right to continue as you are for now, continue trying to heal yourself and repair your life, have the best relationship you can within the limited contact you have with DC and hope for a good relationship with them as adults once they turn 16/18.