I would recommend you to check with school first of how long their process would be as it’s could take time especially during the new year holiday.
If the process of your school are confirm between 1-3 weeks and you are ok with the overstay fee of 500 bth per day. It’s should be best to remain and pay for overstay and school can remain using the current documents they have already prepared for you.
However, if the process is unsure of how many day/week to be able to obtain your new visa due to preparation and approval of visa documents. It’s would be best to leave country and back with visa on arrival or even tourist visa then changes it here in Thailand. This will give you and school enough time at least 4-8 weeks with an extension of maximum 30 days to process all documents/approval necessary
Please be note that you can only apply for visa on arrival or VOA for maximum 4 times per calendar year
However, in general for changes of visa type in Thailand - you would require to have at least 15 days of remaining current visa. Also, non immigrant to non immigrant visa can’t be changes in the country.