From a legal (visa) perspective, the only way(s) to legally teach music in formal education in Korea would be to either have an F-series visa (permanent residency, married to a Korean citizen, have Korean ancestry), or an E-7 visa (foreign expert). International schools in Korea can only hire teachers on E-7 visas, which you might qualify for (you already have the correct degree, your experience and/or licensure would also come into play)
https://www.seoulforeign.org/employment/faqshttps://immikorea.com/en/e-7-visa-foreign-employment/ From a professional point of view, in order to teach at one of Korea's proper international schools, you really should (or must tbh) be a qualified and/or licensed teacher. Your degree in Music Education puts you in this category, though your experience level would also be taken into consideration. Korea doesn't have many international schools, meaning competition for jobs is high...but with the right timing and luck, anything is possible.
Places to look for jobs are:
Search Associates
Schrole
ISS International School Services)
TES
Hiring season typically starts in October/November for the following school year (following September), and lasts till May/June-ish, though most positions would already be gone by February.
I cannot state this enough...DO NOT entertain offers from schools who want you to teach music on an E-2 visa
http://www.koreaherald.com/view.php?ud=20170622000872Good luck!!
