Im willing to bet your kids have a good idea of which parent has been there for them, values them and tried to keep their family together, no matter whose name is first on a divorce affidavit. But if you stay indefinately, you might run the risk of your kids growing remembering that you let your wife run around with someone they have to see at school, everyone in school and town knows it, and you did nothing to stop it.
And if you really want to keep the house, starting the divorce process is the best way to do it. There is probably no other way to force things to change or to get her out.
You are contemplating buying a fixer upper and moving out, but not filing for divorce? Is moving out and spending all of your time fixing up a house and stressing about two mortgages, while your wife stays in your house on your dime and completely beyond your control really any better for your kids?