I read this about three times so I completely 'got this'.
I think that you need to be professional with your ex husband and new gf. She seems like she is trying very very hard to get along with you.
You come across really needy and I believe there WOULD be a reason for you to be seeing your kids 'every other week'. I'm not sure how the court system works in America but at least in my state in Australia it's almost impossible not to get 50/50. If that's what a parent decides.
I also am wondering why you on an infidelity forum when you were trying to be the other woman and clearly have no remorse.
I think counselling may be helpful for you and an extension your children's lives.