Welcome. Know that you did NOTHING to cause her to do this. That is 110% on her. Her inability to communicate effectively is on her. Her crappy choices, yup on her. Do not allow her to make you feel less than a good man because she did this.
Now How to move forward. It sounds like you want to R, and she is willing to give a R a go. However there are some really serious things that have to happen for that to be successful. You think it's all done and over, and it may be, but if it went on for 3 years, I have serious doubts. She absolutely has to send an NC (no contact) letter to him. She has to recommit to you, and ask him to NEVER contact her again and to respect her choice to heal her M and herself.
You also get to drive this ship of R, and you need to make sure you have 100% transparency. Meaning you have access to each and every electronic device, email, and phone record, you have access and can account for every penny she spends. I would also urge you to look for a burner phone, it's been a month, and that is the go to thing for the Wayward spouse to get the second they tell you they are willing to R.
Typically the wayward spouse doesn't want things to end, afterall they have the best of both worlds, the fun and excitement of her A, the financial stability of you. Don't for a second fool yourself into thinking she is all done, and washed her hands of the whole A. It doesn't happen like that.
I also have a few things I recommend all new BS's do when they start down the road to R. You want to make it work for the kids, great, but be strong and show them what a healthy man is, don't allow her to break you and your relationship. Your kids deserve at least one parent that is strong and healthy. For that reason I urge you to see D attorney, even if you aren't planning on it, you need to know how this can/could play out should she choose not to do the hard work, and you need to make your decisions from a place of knowledge not supposition.
Next make yourself a priority. Focus on keeping your mind and body strong and healthy, if you are having trouble with eating, and getting good sleep call your Dr and let them know what is going on. Many folks here benefited from medications in the early days. There is no shame in it, and without proper nutrition and sleep you can't make good sound decisions for you and your children.
You also need to demand she is tested for STD's and do not have unprotected sex with her until she can prove that she has done this and has a clean bill of health. I'm sorry but a "tennis coach" having an A, sounds like she probably isn't the only one, so you are most likely at risk.
This is a long road to travel, and it may seem like I am trying to shoot down any thoughts of R. On the contrary, I want people to R, especially with families, but I also know that it takes a LOT of HARD work from BOTH spouses to do it successfully.
Keep reading, keep posting, look to the library on your left side of your screen, and read the bullseye posts in the Just found Out forum. These will give you some idea of what you are in for, and what is reasonable and not from your WW.
(((and strength))))