Please keep everything together.
https://forum.freeadvice.com/divorce-separation-annulment-36/can-i-ask-wife-leave-house-634502.html
Some of the things considered:
Status Quo - who has been the primary caretaker of the children. <-- This is more important than who is committing adultery.
Best Interest of the Children - this has more to do with providing the level of stability they are used to, rather than who has deeper pockets, or who slings more mud at whom.
I think you should quietly document the care you have been giving the children, and Mom's extracurricular absences. Your documentation may not be admissible, but at the very least it will help you to keep track of the facts as you see them and be consistent.
I think you should refuse to move without the children, and refuse to allow Mom to move out with the children, unless there is a court order.
I think you will have to accept that you will have joint legal custody. The question will come to parenting time.
Expect your wife to want to keep the kids, keep the house, and have you finance everything. Expect that she will not understand that you are hurt, and that your pain is a reasonable reaction. She is in blissful selfish mode right now. She does not care about anything but herself right now and thinks that she has more than earned her happiness. Expect her to continue showing you no respect.
So, let her have her fun. Do everything you can to build up the status quo in your favor. Don't complain about her to anyone but a therapist or secret journal (non-electronic). Your children are watching. Take the moral high road as much as you can in front of them. Realize that although she is neglecting her children for a bed warmer, accusing her of neglect will get you nowhere. Unless she does something that would get her an indicated report with CPS, do not even breathe the word "unfit". (And really, if your spouse does get that, it's not something you "happy dance" about.) Let you wife to do all the damage to herself.
Be patient. Focus on the kids.
Oh, and don't consider dating until the divorce has been finalized. You're enough of an emotional mess during this that your children don't need your attention further divided.