Okay, I do not know what the arrangement is... You say 'she', but I am assuming that your Ex will be taking them, along with his girlfriend, to visit her family. She would have no right to take them anywhere on her own.
I can certainly understand how you feel!!!
However, the answer is not really simple.
1. What are the custody/visitation arrangements? Is it allowed for your Ex to take the kids out of state without your prior approval? When you mention Tennessee, I am assuming that you do not live in Tennessee. How far of a trip is it?
2. I am assuming that it is indeed perfectly acceptable for either you or your Ex to find a Significant Other. Yes, this may not be really 'comfortable' at first. But, really, if she and your Ex have been together almost a year now, as much as you may hate it, she is probably now a part of your Ex's and his children's (your children's) life. It may be time to make an effort to make it more 'comfortable'.
Whatever happens, and whatever you decide. Unless there are issues regarding the children's physical safety and restrictions on your Ex's custody, it sounds like you might have to compromise. After all, he is their dad.