Welcome to the club no one ever plans to join, but are glad they found.
It seems that you have been doing a fair amount of reading here, and that will help you figure out what to do, and how best to go about saving your sanity.
There are a few things that you need to do though, and these are to protect yourself, and your kids. The harsh reality of the situation is you are in a current state of sharing your spouse, if you haven't already lost him. Can you get him back? Only if he chooses to remove his head from his rectum. YOU CANNOT MAKE HIM DO ANYTHING. No of us has the power to force someone to do something they don't want to. We do however have the power to make life really uncomfortable for them if they continue to make choices that hurt us.
The following is a list of To Dos that I give nearly every new member:
1. Go see a lawyer, not next week, not next month, but NOW. Get the information of how this will go down should you S or D. Find out what your rights are, and what his responsibilities are. Prepare yourself with knowledge. Knowledge is power, and that will give you strength.
2. Go get STD tested, and do not engage in intimacy with your spouse until he too has been tested and provides you with written proof of a clean bill of health. This is where he starts to get uncomfortable.
If you are having trouble with the basics in life, like eating, sleeping, and staying hydrated, then by all means talk to your Dr about this as well. Many of us benefited from antianxiety meds, and antidepressants to get through this. Given your stated history it is even more important that you stay on top of this.
3. Read up on 180 and hit it hard. This is to protect yourself, and help you gain inner strength, and perspective on your relationship. This NOT to win him back, for many here it is an added benefit (This will also start to make life really uncomfortable for him).
4. Tell him what you require for R, and that should include complete transparency, an NC note that he writes, and you get to approve of and watch him send it. If he balks at letting you see his phone, you should be wary that something is still ongoing. If he balks at letting you on his computers/tablet whatever be concerned. If he willing gives you this be suspect that he has a burner phone.
You will see the phrase "Trust but verify" here, a lot. I disagree. I think it is foolish to trust. My version of this is "Do NOT trust, verify verify verify, and trust will slowly rebuild itself each time you find nothing".
Along with transparency you have to out this to OW's partner/spouse if she has one. DO NOT tell your H you are doing this, just do it. A's thrive in secrecy, and day in daylight.
Be very clear, concise, and prepared with consequences should he not be willing to meet or follow what you need for R to occur.
5. Know and tell yourself this every day. "I did nothing to cause this. He made his own choices. I am strong, smart, and capable. I will survive this and be stronger, smarter and fierce in the end. I deserve much more, and I am going to demand it from now on."
You of course can put your own spin on this, but you do need some sort of positive self talk. It helps you to find you again.
6. Do one nice thing for you everyday. You deserve it. Take a long bath, get your nails done, go see a movie, whatever but do it because you enjoy it and you deserve it.
From this point on you determine how this goes. IT's up to you, you can choose to attempt R, you can choose to D. Whatever, you want. But know this. You will not have resolution to it for quite some time. It will be a rough road for a while but in the end you will be ok, if you act with integrity, and demand the respect you deserve.