For me infidelity is a deal breaker. Infidelity has nothing to do with him drinking. So there are more issues here than just alcoholism (which is enough in itself). Additionally, when infidelity and alcoholism invade a marriage the only way to rebuild is through transparency. Impossible to do when you are seeing each other 4 days or so a month.
"He is begging for a final chance to go into recovery and prove himself" . translation: "I will go into recovery in order to keep you from divorcing me". This is not going to work. If he was serious about recovery he would be there RIGHT NOW. He isn't serious, he is just manipulating you.
How would I handle? Tell him I need some space and want to enjoy the holidays. Lets talk about it after the new year. New year comes, new life, new me and good bye.
Again, there would be no second chance after the dating sites and texts etc. others may feel differently. BTW how weird that he deleted his part of the conversation, but not hers.....almost seems he wants you to see it. My guess is you only know a little about what he is really up to.
Sorry you are going through all this and welcome. Hope we can help you out.