As far as the legalities go that is different in any state. Here in IL you need the other parents permission to move out of state otherwise it's a no no. At least in the cases I have been a witness too.

On the topic, I am sorry that your hurting and it's comendable that you want to save your marriage. I say go to counseling, show her by action you are trying to be a better man BUT don't let her bully you out of guilt. I have been cheated on (Not by LK, pre cat days) and it sucked. Ruined my confidence in myself. She's asking why wasn't she enough, why wasn't she pretty enough, what did she do wrong, how did I not know, how could he do this to me. Those are some of the things running through her head. So I say to prove to her that you are willing to do whatever it takes to gain her trust back
1. Go to a Dr, get a clean bill of health. No STD's to ease her mind of that,
2. Start seeing a marriage therapist on your own. If she see's you putting in the effort she may try too.
3. Take ownership of what you did. If the kids ask in front of both of you say something along the lines of "daddy did something really naughty and he's in kinda like a time out because he has to learn from his mistakes like you guys do"
4. DOn't push but when you DO speak, i assume you do about your children, tell her I Love you and I am trying to become the husband you deserve. Not over and over, just once during the conversation to reiterate to her that your trying.
5. ABSOLUTELY NO CONTACT with other female EVER AGAIN. Change jobs if thats what it was, change health clubs, whatever. Do not put yourself in the vicinity of the woman again, for your wifes sake.
It's gonna take time for her to decide if she can forgive you, give her that time and space, be the loving husband when she allows you to. ie when you pick up the kids open doors for her, get her coffee or soda or whatever while you are up. In time she will either forgive or file for divorce. If thats the case you know what you have to do in your next relationship