Yes, it is possible for the pain to move around with TMJ, from side to side, from tooth to tooth. That's part of what makes it such a difficult disorder to treat.
the problem with just fixing your teeth and seeing how it goes is that you are spending all that money to take a shot in the dark. If you know what the problem is before beginning treatment you can treat the problem correctly. Just taking a shot in the dark can lead to further problems because they are guessing, trying to treat a problem they are not sure of. I personally would not allow treatment to begin on me unless I was certain of what the problem is. You could end up spending all kinds of money then later learn you had a displaced disc and they were not treating for that so you could end up with more problems than you have now. It's always best to know what the problem is before you start treating it.