If he just broke up with gf three days before, he'd probably been carrying on an EA with the long-distance coworker for a while before.
As for NG - if he can't man up, dump him. Sorry to sound so harsh, but you're right. You have three kids to mother, you don't need another one. It's one thing if he complains about his work but is carrying out a proactive plan to fix the problem. But if he's just a guy who's gonna whine about it but not do the things necessary to change the situation? Bye bye.
You need an adult male.
My friend was dating a guy, she's a musician and he was a musician in a somewhat known band from the 90's. He's in his fifties, works in a record store, can only afford to live in LA because his landlady hasn't raised the rent, has thousands of dollars in credit card debts and drives an old clunker. He loved to cry "poor me" but never did a single thing to better his financial position while she was with him. She dumped him because of this (she's a widow with a son to raise and like you, didn't need another kid). Four months later he was "in love " on Facebook with a new woman and proposed to her after a couple of months. She appears to hav a solid middle class job and we can see that she must be rescuing him financially. They're a long distance couple and she is almost always the one to come to him. It's been a year but no date has been set and scuttlebutt is that she keeps offering to bring some of his stuff home with her and he keeps declining.
We're pretty sure he doesn't really want to follow through, just wanted a rescuer. I feel bad for her, I imagine being swept off her feet but a rock star has kept her from seeing who he really is.