How can I cancel my old visa and work permit while applying for a new job in a different province in Thailand?

IF you're on a visa, you're stuck you can't cancel a visa and get another one inside the country BUT if you're on an extension of stay (LOOK at your current stamp) you can.

You get documentation from your current employer stating you will stop work on xxx date, you go cancel your current work permit, then go to the immigration office a couple weeks before the termination date on the paperwork from the employer, they cancel your current extension and stamp you in until the termination date on the paperwork.

THEN you get down to the new immigration office and apply for a new extension with paperwork from the new employer (and apply for a new work permit too)

You need to be coordinating this with BOTH your current and your new employer AND BOTH immigration offices. (As in you should make sure that the new immigration office is going to not have any problems with the way I just outlined you should do it)