OP, I reckon you have to let this go. They didn't shag, they didn't even kiss. She was in the shit, he gave her some cash. You found out and went mad, he left his job and there hasn't been a peep between them since.
If what he's saying is true, then he seems like a decent fella and you should be proud of him, not trying to catch him out and test him and still be so full of anger about this. Okay, he didn't tell you he'd given her money, but I'm not surprised he didn't tell you if this is how you've reacted. Taking her for a coffee at lunchtime when she's upset isn't a red flag, to me. To me, it's a gentleman move -- maybe he was playing the 'white knight', but that's not a bad thing, and it doesn't mean he wants to shag her. He sounds like a decent fella. Not a twat. So maybe stop treating him like one?
If he was a male colleague, would you be reacting in the same way? Would you say they were having an 'emotional affair', that he' 'betrayed you', all that? I doubt you'd give it a minute's head-space.
People spend a lot of time together when they're colleagues. They tell each other about their home lives, their children, they share stuff, they get close. It's great when work is like that. Feels like family. Okay, sometimes it gets a bit deep, but you have to trust your other half to do the right thing, and it doesn't sound to me like he's done bad.
You've been together for years. You say he's a good husband and a good dad. I wouldn't want you to throw all that in the bin for nothing.
Give him, and yourself, a break. Stop winding yourself up. Unless you're looking for an excuse to bin him, I'd stop all this wondering and make a decision. You either trust him, or you don't.