BTW, if my wife was f**king someone behind my back, she would be the one moving out. Nobody betrays me and then kicks me out of the house. If she's so hot for some other dude, then SHE can be the one who moves out.
Seriously, if she doesn't want to be around you then make her leave. Why do you have to be the one to move? Why do you give a rats bellybutton about whether she wants you around or not? She's the one who's leaving you, so let her be the one to leave the house. Her forcing you to move out is just her manipulating you into leaving her, so she can justify the breakup in her mind as something that YOU wanted.
I'm not kidding dude. It's your house too, and you have every legal, logical, and emotional right to stay there. She's the betrayer so she should be the one who gets out. If she doesn't want to uproot the kids, they can stay, but she really should be the one who leaves. Don't let her BS you about the welfare of the kids. She already demonstrated she doesn't give a damn about you or the kids, and she's just using that excuse to get a free house. Make her be the one to make the decision.
No kidding, I don't think you should leave. If she wants to leave, fine, but don't let some cheating potato kick you out of your house just because she decided she likes someone else better than you. That's not your problem, it's hers, so let her deal with it.