@SArah **********
Honestly, I'd make a plan B:
1. You're only allowed to work at five schools, OR have a temporary teaching license for 5 years (the last I checked).
2. The Thai government does *NOT* want to hire teachers inside Thailand; they want to hire overseas now. See
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51/govt-to-be-stricter-over-work-permits-for-foreign-teachers
3. IF you're still on the original Non-B visa (NOT an extension of stay), and your former employer is written on your Non-B visa, they may want you to get a new visa from outside the country.
However, you are working at a government university, which may be different than a primary or secondary school