I wish you'd mentioned all this before- it totally changes the whole picture.
But- your Dh is enormously damaged: he was beaten as a child, there is dependency in the family ( on drink) he has all the guilt that Catholics often have.
Is it any wonder his head is totally fucked up?
He has HUGE issues which need professinal help. Anger is one of them.
I'm not a counsellor but have 3 very close friends who are- so we talk ( not about clients of course) but about issues which affect people.
Psychodynamic counselling takes people back to their childhood- and it's there that the foundations are laid for their adult personality and behaviour.
I would bet my last pound that your DH has excrutiatingly low self esteem, that he was "abused" as a child by a violent father, that he doesn't like himself, and that this OW ( she is almost an irrelevance) is just a distraction and ego boost.
If his childhood was also devoid of real love and affection, what role model has he had to help him be loving to you? If his parents love was conditional and not freely given- how is that affecting him now?
This is a huge minefield.
TBH I wouldn't waste any time thinking about the OW- she's not important- he's hardly a good catch is he at the moment?
What is missing from your marriage is true communication.
Could you make his seeking help a condition of staying together?
Does he say he loves you? Does he say he wants things to work?
Finally, you could see your GP and express your worries and ask if they could refer your DH for support- they have a duty to him to keep confidentiality but there is no harm in anyone living with a depressive consulting their GP ( assuming you have the same one?)