This is a very tough situation to be in. I'm sorry for how much pain you must feel. Basically, the marriage is already over. She is not committed to you. I don't see how you could ever trust her again.

In the middle of this are 3 children, which complicates things immensely. They're still young. You do have the option of simply staying there and trying to emotionally detach from your wife. That keeps you near your kids. It eliminates you needing to set up a second household. But it keeps you on the receiving end of a lot of pain. Moving out, however, does not necessarily reduce your pain. So you have a choice between 2 bad options. Your wife may actually want you to stay because you being there makes it easier to manage the kids. Only you can decide if that is a tolerable option. You don't have to be in a hurry to go with the option of leaving. Once you move out, you'll probably be locked into that option.

My strong suspicion is that your wife has emotional problems. I don't think therapy can bring about any quick fixes. She also has a character problem. Character goes way deep. Besides betraying you, she lacks respect for herself. What kind of guy hops into bed with a married woman with 3 kids? These guys she has affairs with are not great guys. That's why they are available to mess around with her. Her life is headed in a bad direction. You could choose to stay where you are and see what develops. Eventually, she will get hurt. That might make her appreciate you more. But I wouldn't count on that either.

If you want a healthy, loving relationship, you're probably not going to get it with her. Divorce could free you to find someone else. But that's no guarantee either. Another option is to carry on with an "open" marriage, where you see other women. But what kind of women will be available and up for that? I'm not seeing an obvious best choice for you. For now, you might want to stay close to those kids and put her out of your mind as best you can.