Although I am a little late answering your inquiry I would like to let you know that it is very much possible for you to acquire a non-teaching job in South Korea and Japan.
As a native English speaker, teaching opportunities are indeed abundant however, jobs in fields such as Media (written, digital, television) , big corporations (Both Korean and foreign) as well as Tourism are available if you know where to look.
I lived in Korea both as a student and an employee back in 2013, and had American friends who held non-teaching full-time jobs who had attended college either in Korea or the U.S.
The only point I would like to highlight is that mastery of some level (medium to high proficiency) of the Korean is highly recommended as it would prove to be quite difficult to get a non-teaching jobs without it.
The best plan of action would be to start looking for opportunities online. There are quite a few webpages that offer listings for available Jobs in Seoul and other major cities such as:
Craigslist: seoul.craigslist.co.kr Worknplay: http://www.worknplay.co.kr Kopra: https://www.kopra.org/
Another great approach would be to look into academic exchange opportunities if you are currently enrolled in school, and/or the various cultural and language programs offered by educational institutions in Korea => for more info https://www.studyinkorea.go.kr/
Visiting Korea in such a context before settling down and job hunting will provide you with a much clearer perspective vis-a-vis the working culture and employment opportunities.
Best of luck!