bv5913, here's my experience with the ex "moving away with the kids":
After I was divorced from my first wife back in the 80's, she remarried and wanted to move out of state with my daughters - from California all the way to the East Coast of the US.
She had to get my signed permission to do that, and I had an awful time coming to a decision. I finally agreed, trying to think of everyone's best interests, but later I had many, many regrets.
If your wife is already using the children to manipulate you, consider that the only leverage you have is not signing that document. Find free legal aid, come up with a plan, and negotiate hard for what you want. Assert your rights!
In the years to come, you will be glad you asked for the best possible terms...