I am so sorry to hear that you are in this position OP.
Many of us have been where you are, and it is absolutely devastating, so sending you massive hugs.
I came home from my Mums when my son was 6 weeks old to find my partner at the time (his Dad) had packed his stuff and left. Wasn't answering his phone, messages, nothing. Later that night, I got through, and a woman said 'he's with me now' and cut me off. I was devastated at the time.
I am now a single Mum to 4 boys, and it is hard as I don't have any support (I don't have any family, my only friend lives two hours away) but I am an amazing Mum and now the kids are getting older, they appreciate that.
Firstly, be kind to yourself! An easy tea one night when you are tired? That's perfectly fine. Tired at 8pm when the kids are all settled? No problem... get the rest you need. You can do this... and smash it, even though you don't know how at the moment. You will find the strength to for your babies I promise you.
Secondly, and quickly, make sure financially you are ok. Anything you have with him, make sure you protect yourself and take copies of anything relevant that you may need later on down the line (any documents, payslips, proof of savings etc). Sounds like the last thing you need, but it will help you down the line to have copies of these things if required.
Try not to focus on him and the OW and focus on you and the kids, taking one day at a time, as you will just devastate yourself further if you focus on them.
Could contact be done with his family instead of at your home? It would enable you the break that you will need, and give you time to sort out the things you need to.
Also, see if there is a childminder or likewise nearby that has a DBS check that you can use on the odd occasion if needed to have a few hours to yourself.
My youngest goes to a childminder who has been babysitting for me since I moved to a new area 3 years ago. She is amazing and it gives me a couple of hours child free when needed to just breathe, and with her profession and DBS check, I knew they were safe.
The best revenge is to show him that you don't need him, and you are the best parent to the children that you can be 💕