STBX made the statement that she would not stop me from moving with the boys to another state if I had a new job offer. Well then why is she fighting for custody?
I doubt this to be true if it actually happened. Because if it was, she would just leave.
She's saying a lot of things. She's re-writing history. Don't believe what she says. Watch what she does.
When do you move into the new place?
I agree that strengthening your validation skills will be helpful. Reflect back to her, without agreeing or disagreeing. Wonka has a great post with wonderful phrases to use in the validation cheat sheet thread.
A few examples: -"I am gonna have to think about that a little more" -"I am sorry that you feel that way" -"I can understand why you would be upset"