@Eduard ******************

You do not need a new visa. Your new school needs to get the letter of approval from the previous school allowing you to switch the extension of stay. They clearly don't know the process/ Lazy/ Do not have the means/ Or they are in a different province. Also, an extension can only be done any day before it expires not on the last day provided there is 15 days remaining on it. I don't know what stories they are telling you.

I dunno what immigration you went to, but if you have all documents ready to go, Chaengwattana allows it in 15 days.

One needs 15 days left on the current extension to get it transferred to another from another employer. You clearly have 15 days. If you can get the paperwork and a job in that time.