Hi Nigella. I have been lurking on this thread with some interest and concern..partially because I have some experience of some of what you are going through
I found out three years ago that my OH had been contacting, taking for dinner, fantasising about a work colleague for around six months. He told me that if his world was different (ie not married to me) he would want to be with her. I am convinced that had I not discovered through looking at texts it would have continued to this day. He still works with her and I think the EA with her was only fully broken last year
I don't think shouts of LTB helped me or will help you. I am still with him because I love him no more no less. I am not a weak, feeble and needy woman, in fact I am now stronger than I have been at any time in our relationship. I made a choice to stay..but that came at some cost. I will never fully trust him again with my feelings and that is a hard way to live. He has made mistakes and he is a liar, and still makes mistakes and lies. That is something that you will probably have to face too if you stay
At the same time I sought out legal advice and know exactly where I would stand if we divorced. I live in an odd but loving relationship that is without delusion for me but with a love that in its own way nurtures me, the children and him and I am, for the moment happy with that. If things change then so be it I have the tools to deal with that as and when.
In my experience I doubt it has been a just few coffees etc and if you peel the onion there will probably be more. However, it is your choice how much you want to know and how you want to deal with it.
I hope this helps. Take care