Not to butt in but I have a couple of thoughts for you to think about. I suspect I know what you will say or feel in response to some of it.
First I agree LRT is what you should be doing. Secondly, you have already stated your boundary by telling H he had to choose. Stand firm in that. Meaning, while you may not be ready for D you are also not willing to play second fiddle. So until OW is out of the picture no relationship with you.
Thirdly do you really want to be in a M where your H doesnt wholeheartedly want to be in or be committed to?
Finally- it is up to you to really decide if OW is a deal breaker for you. I suspect not but honestly the LRT is your best option. The purpose is two fold - one to protect yourself from further hurt and two to REALLY give your H a taste of life without you and your family.

As for the kids I'm not sure of their ages but yes if you guys are separating you should be honest. How much detail you go into is up to you. My experience is that affairs have a tendency to wither when exposed publicly. That said you do not want to just be telling everyone and your kids. At the same time if someone asks don't lie either. For me I have always felt that natural consequences work best. That way they feel real emotions for their actions and repercussion but also it keeps you from being blamed for those consequences.

At this point, I would LRT completely. Once you feel a little stronger perhaps talk to H and figure out when and how to tell your children if that is the step you will be taking.
Someone said earlier it is all about your him and they are correct BUT you need to focus on you!! Make it about you. Examine why H felt need for affair in the first place. There's a book called after the affair. I highly recommend it.

The ball is in your court. I know as counterintuitive as it seems you are not powerless here. Take control of your life and let H control his.

You are going to hear it over and over but detach. It is simply a way of protecting yourself!! YOU MATTER!