What is the process for obtaining a new extension of stay and work permit in Thailand while changing jobs?

You get paperwork from your current employer stating you stop work on xx date, and you go to the immigration office a week or so BEFORE that date to cancel the extension. They'll stamp you in until the date on the termination paperwork and that should give you time to get to the other province use the documentation from your new employer and apply for a new extension.

You need to be coordinating this with your current (soon to be former) and your new employer AND both immigration offices. It CAN be done you just need to get your shit together to make it happen.