I agree about contacting an attorney who understands your options, the likelihood of achieving your primary goals (full custody), and how best to proceed with a cheating spouse who refuses to vacate the house.
My ex-WW (no kids) cheated with a coworker and moved out of our rental immediately. Took her quite some time to get out of the fog (still not totally clear to me if she is fully .. 5yrs later). About 1yr after D-Day, I told her she needed to file and she did. We've been divorced nearly 4yrs now, but she often reached out to me following divorce. It seemed she wanted me to do the pick me dance (and I did early on, but slowly pulled away with time), as she would never recommit fully to reconciliation, therapy, etc, but wanted me to stay in her life. I know I could never be just her friend, and made that clear. We'd go weeks, months, one time almost a full year with NC (because I stopped initiating).
She's coming to visit me tomorrow .. as I moved out of state over a year ago and have tried not to look back. I've dated, I've focused on me, I've done my best to move on. I haven't been able to change what my heart has always wanted (even after meeting some amazing women).
We aren't close to being in a place of reconciling, but it feels to me like this could very well be the last time I open that door with her. There's still so much love between us, but I know I can't keep living my life like this, and have told her I can't be doing this or still be in this place 5 more years from now.
Divorce doesn't end things .. especially if you have children. You pursuing divorce does stop the craziness you're living .. the limbo hell you're stuck in now. You need to speak to an attorney to find out what your choices are and how best to proceed for you and your children. You need to think and focus on your crazy wife (you cannot control anything she does right now), and focus more of your attention on what you can control (filing for divorce, moving, selling the house, etc). For me, as soon as I began planning my life independently from my ex-WW, the craziness began to subside, the pain began to subside, and I began to pick up the pieces of my broken life and heal.
Best of luck.