There you go again worrying about the future and preoccupying your mind with matter of which you have no control of.
Yes, your wife checked out, yes she is going clubbing, but that is not your concern and none of your business.
And WTF regarding your children's future is bleak?!? You are their father and you take care of them, why should their future be bleak?!? Have you seen children in Somaila, Syria, Iraq? How dare you call your children's future bleak?
And again you go with the mind reading... Assumption is the mother of all fcukups... Don't do the mind reading. Really! No good can come out of it. I get you, dude, I reall do, but you are just burying yourself in self pity and you are doing yourself no favors.
Holding her hands and telling her you are sorry will not sway her. We have all told you that and yet you refuse to believe. So my suggestion is go ahead and do it. Grab her hand in the kitchen and spill your soul to her. I know you have to get it out of your system, I know the uncertainty is eating you inside. So, do it, get it out of your system and be done with it.
You are doing quite well for the stage you are in (being 4 months post BD), sure you could be doing a lot better, but you will get there.