What is the process for changing employers on a Non-B visa in Thailand?

Hello! I need some clarification on the process of changing employers/non-b visa/work permit. I work for a public school on a non b visa with a work permit, and I want to resign and find a new job. I have to give my current employer 30 days notice. My question is about the process after getting a job offer and giving 30 days notice. Will the new employer be able to process paperwork in these 30 days whilst I'm still on the non-b with the current employer (like transferring the visa over to the new employer) or will the process start over once I'm officially resigned and the work permit cancelled. Most info points to doing a border run, does that mean a border run to get a tourist visa and then again to get a non-b for the new employer?