Release Letter is only needed in below cases.
1)You change workplace from Hagwon to public school, in this case employment period that you have worked doesn't matter.
2)You change workplace from Hagwon to Hagwon, in this case employment period that you have worked has to be at least thirds fourth.
3)You change workplace from public school to Hagwon, in this case too your employment period that you have worked has to be at least thirds fourth.
There is no fixed form as long as the release letter contains your personal information, current workplace name and employment period and school signature called " Dojang", and the workplace name that you will work for and it can be written in Korean or English.
If your situation doesn't fall under above cases, you must terminate current visa ( Your visa termination can be only done by your employer ) and apply for new E2 visa sponsored by new employer. It means you need to submit your visa application document again ( No need release letter and if you apply for new visa without leaving Korea ever once, you don't have to resubmit your criminal record)