Since you're not a catechumen, there's no need to provide references. But, once you make that decision, it is important to stay in one place. Main reason being that the priest is responsible for giving Holy Communion and he needs to know the person. It's also an important relationship. May I ask what your job is? Is it something that brings you back around to the previous places? At least the priests get to see you again.
I'd have to disagree on the not providing references if one is not a catechumen. There's a bit of difference between someone who has just been attending for a few months and hasn't read much and someone who has been around for 5-6 months and is perhaps ready to become a catechumen.
It would make sense for Dhyn's old priest to say to the new priest, "Dhyn has been attending for X number of months and has read X, he's a regular at Liturgy and is possibly ready to become a catechumen."
As for the staying in one place, it sounds like that is not possible with the OP's employment situation. If your job requires you to move, it requires you to move.