I think the only way you can make her leave is file for Divorce from Bed and Board, and that takes a while. She is carrying on a blantent affair in front of you AND the kids with no remorse at all? I’m sorry. The fact that she doesn’t work places her as the dependent spouse in NC. You work 3rd shift, so childcare is an issue. You are in a tight spot, I’m afraid.
I would consult a lawyer and see what he/she advises. Start documenting times she is away. If you have any proof, start making a folder. If she is so blase about the affair, she’ll leave evidence behind. Take it and keep it. Maybe actual action on your part to address this issue will prompt some resolution. If you talk talk talk and do no action, she has NO reason to change the way she’s doing things now. You’ll have to be pro-active. Good Luck.