Hi, Happy New Year. I don't think you are being unreasonable. Part of the responsibility of caring for children's emotions during family break up, is in my opinion, letting children know that they are safe and protected. Its valuable to let them know that some behaviour is unacceptable, without spelling it out to them. If you only just separated in September I think its appalling that the father should be introducing a g/f so soon. Remember they may have a history but the children are not aware of it. The whole thing of not paying child support is disrespectful to the children. I've no idea of your legal position, american family law strikes me as insane. Feel all your feelings, keep on an even keel for the children's sake, they will remember what you have done for them, which of you was there for them. In my own experience a child generally loves the dysfunctional parent too.
its late, i can't flip back to your message to get more detail w/out losing this post. Time is a healer but sometimes you think you've got it sorted then it all comes flooding back. If you can don't get drawn into any conflict but stand your ground if its reasonable. And even if its some kind of hurt/messy/jealous/confused standpoint your coming from you're allowed to feel those things, your human. He'll be a dad if he wants to, thats up to him. One day at a time,
much love MM