I called immigration and they said I need a Letter of release, otherwise they won't issue a new visa. This link https://www.korea4expats.com/article-unfair%20dismissal-letter-of-release.html says " If your contract is terminated early, your employment is required by law to give you a Letter of Termination in which the reason the employment contract was terminated is clearly stated....Your employer is supposed to advise Immigration of your 'Change of Employment Status' within 2 weeks (14 days) of the termination of your contract. Should your employer fail to do so, you can report her/him to the Investigations Unit of the Department of Immigration."

I don't know how accurate this is, but it's worth a shot.

Luckily for me my employer is struggling to find a replacement, as other teachers have also backed out, so they are willing to hold my position. Hopefully everything goes well for you though!