you will tell her straight & simple, if she isn't concerning herself with the kid's welfare and is just concerning herself with thoughts of the OM, you will go for full custody, her kids better be her main priority and if she can agree to that tell her than you will then go for shared/joint custody - NO weekend dad crap, be clear of that so there is no mistake and no assumptions. Yes ruffle the feathers a little bit, she expects to run you over because she's been used to doing that for quite some time.
Tell her that you aren't the same person you used to be, you won't just accept what she gives you anymore. Those are your kids just as much as hers and you won't be relegated to the position of "weekend dad", they will be in your life and you will actively be in there lives.
You can also tell her that you've been monitoring & recording everything and you've been speaking with lawyers - she can be sure that YOU won't be taken advantage of anymore.
Tell her to enjoy her life with the OM, you will be enjoying your single life as well.
As far as the kids are concerned, she doesn't get to take them out of the home until she has her own place. Living with the OM is not an option for your kids, you've spoken to a lawyer already and confirmed as much.
Wake her up, right now she thinks she's running the show, time to take back some control.