If somebody wanted to could they openly just get up and move to a different state

Anybody can move ANYWHERE, but they have to either have the paperwork finished or at least started. Anybody who drives needs a license for that state (non-drivers some kind of state I.D.) with a correct and current address. Your last utility bills from the old address need to be paid in full, and your new address has to be noted with the gas and electric companies. If you have a paper delivered, notify them of that change.

I had to do all this when I moved from my childhood home to this apartment.