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The easiest way for your sister to accomplish her goal of moving back to NC would be with the ex-husband's approval in writing and notarized as an amendment to any marital settlement agreement that they currently have.
Should that not be an option, then she would need to go to court, get in front of a judge, and see if her request is granted. it should not be much of an issue considering that the ex-husband has already moved out of Texas to Seattle. Just the same as he travels to Texas, he can travel to NC. I don't see a judge disagreeing with her moving away, especially if it will not substantially change the circumstances for the father and would better the mother's situation who has primary custody.
Whether her ex cheated or not is irrelevant in this situation and wouldn't really come into play.
When it comes to children, it is definitely worth the fight.
Also, if she is making less money, she can apply for a recalculation of child support payments.