I was divorced in Arizona and wanted to leave the state as well. In order to leave, you must prove to the judge that it is in the "bests interests of the children". If your stbx is a good father and will provide for the kids, your chances are slim.
In my case, my x was addicted to drugs as well as abusive. Had to prove it through drug testing. Also, ex refused to pay CS, did not pay mortgage, made false accusations against me. I was a stay at home mother with no income. My family was all back east, I had no support system in Arizona.
Judge ordered family evaluation. X threatened the evaluators (when confronted with the positive drug tests) and me during evaluation. This proved the abuse.
Anyway, Family Eval and GAL both agreed I should have sole physical and legal custody and should be allowed to move back to my home state. Judge signed the order and I left the state within 48 hours.
This entire process took 9 months. It was long and nerve-wracking but worth it.
As an aside, ex hasn't paid support since we split. The only exception is when he's arrested and has to pay to get out of jail. He hasn't seen his kids in 5 years. Hasn't talked to them in about 3 years. No b'day cards or gifts, no X-mas cards, no letters, nothing. My kids are better off without him. So glad the judge let us leave AZ.