I'd suggest that you avoid using X to teach morality lessons whenever possible. Seriously, how often did you use the other parent as an example of bad behavior to teach your kids lessons when you were married? Can you imagine: "Oh, see how bad daddy is calling in to work sick when he's fine. He's lying and cheating the company. Bad, bad daddy. Don't you be like that when you grow up." Who does this help? There is no reason to tear down your kids' father simply because you are no longer married to him. Your children and their R with their father and the level of comfort and safety they feel loving their father is fragile right now. Don't make it harder on them by beating up on their dad to prove a point, no matter how good it would feel.
As for the latest mess in general, sorry to hear about your pain. It sounds like you were looking for something out of your new contact with X that he was not looking for. That hurts. I'm sorry. But there is nothing immoral about dating.
Lots of people date, X is free to date. And, X is free to keep his dating life private from you. I certainly don't share details of my private life with my X.
Unless X told you that he was once again committed to an exclusive and monogomous R with you, then his new GF is not an OW and his R with her does not constitute another betrayal.
He showed poor judgment in not sharing the fact that she would be present on an outing with the kids. Insofar as he lied, that is certainly not attractive. But it is not illegal. He was probably motivated to avoid your reaction about the GF going on the outing. Not saying this is a good reason, just talking about his motivation. If you recognize this, you can improve your co-parenting R by being less reactive. Again, detach.
He needs to have his own R with the kids, it is not yours to manage. So, stay out of it unless something is legally actionable.
If you must express yourself in an effort to try to control X's behavior, perhaps something like this will not do too much harm: "I believe that it will be best if we can be honest with respect to when our children will interact with BFs or GFs in our lives. I also believe contact should be excluded unless it is clear that there will be a serious LTR."
It is also unlikely to do much good. X will do what he wants when he wants. X is no doubt trying to be a good father. Sadly, his choices aren't your business unless something actionable arises.
