You certainly could be starting over once you move.I think OP is asking if she has to pay orthodontist 1 the full 18 months treatment if she moves in month 10 of treatment.
Honestly I have the same question. We are military, and I have 2 kids that need braces. What if I move in the middle? Do I pay twice for once set of braces if I move?
Not every orthodontist does things the same way. When I had 2 ready for braces I got 4 consultations on each. Each one was very different. Not just in price, but in treatment. Some wanted to pull teeth, others did not. One wanted to wait a couple more years, the other wanted to start right away. One wanted to make an appliance. There was phase 1/2 or do them all at once.
It was dizzying. I talked to my kids' dentist about it. She said their was no right or wrong way, just different methods to achieve the same results. She helped me pick, and we went on our way.
My payment contract has nothing to do with months of treatment. Rather, it's just a payment plan. I owe $3000. I could pay it all up front or installments. ($200 for 15 months) Even if I leave I may still owe the $3000 for the course of treatment he started. (Possibly he gives a refund?)
If I were to change orthodontists mid stream, I'd probably shop around for someone who would most likely finish what #1 started. But it wouldn't be as simple as finding a random ortho in the new town to take over for 7 months.
Your new ortho is going to prescribe their own course of treatment. Theoretically, it will cost less than starting from scratch, because some improvements were made at orthodontist #1.
Finally how "perfectionist" do you want to be? These days almost all children (who can afford it) get braces. It's no longer for severe cases. Much of it is cosmetic. An ortho can go all out, or not. And often price dictates that.