How can I transfer my non-B visa from one school to another in Thailand?

𝗜 𝘂𝘀𝘂𝗮𝗹𝗹𝘆 𝗲𝘅𝘁𝗲𝗻𝗱 𝗺𝘆 𝗻𝗼𝗻-𝗕 𝘃𝗶𝘀𝗮 𝘄𝗵𝗲𝗻 𝘁𝗵𝗲 𝘀𝗰𝗵𝗼𝗼𝗹 𝗲𝘅𝘁𝗲𝗻𝗱𝘀 𝗺𝘆 𝗰𝗼𝗻𝘁𝗿𝗮𝗰𝘁 𝗯𝘂𝘁 𝗶𝘁 𝗶𝘀 𝗮𝗻 𝗮𝗰𝘁𝘂𝗮𝗹 𝗻𝗼𝗻𝗕 𝗩𝗶𝘀𝗮. <- that would lead me to believe you are indeed on an 'extension of stay' (y)

Which is a good thing (y)

That means you could follow what LLoyd recommended, get documentation from your current employer stating you stop working on xxx date, go to your immigration office in Chonburi a week or so before that date, cancel the extension you're on (they'll stamp you until the termination date.

THEN

you would high tail it to your new immigration office in Ayutthaya with documentation from your new employer and immediately apply for a new "extension of stay".

It most definitely CAN be done, but it takes both employers to work with you to get the documentation you need to cancel the current extension you're on now at the immigration office in Chonburi and then to apply for a new extension with the documents from the new employer at the immigration office in Ayutthaya

If however you are on an extension of stay and you get your ducks in a row, you could cancel your extension ahead of time, and then hopefully your new school would have all your paperwork ready to apply for a new extension (on the day your old extension expires) at your new office, (and of course work permit).

First off are you still on the actual VISA or an extension of stay?