Welcome to the site, tjg_marantz!
It looks like you will have a few options. For non-teaching jobs, check out these sites:
http://www.jobsinjapan.com/ http://www.daijob.com/dj/en/index.html http://www.japanjoblink.com/japan/j_joblist.asp
You could also try private tutoring in French or English or both. I'm not sure how the pay is in Japan but it would be worth checking out.
Private tutoring is illegal in Korea but very profitable. Many people do it and most never have any problems. In Korea this might be your only option. You could try to get some editing work but those jobs are hard to find. There are a few French Institues where you could ask about work/tutoring. here is a link to help you get started with that :
http://www.aftaejon.co.kr/alliance.html other than that, there is not that much non-teaching work in korea, except in factories.
Let us know how it goes. I am personally very interested in the possibilities of Japan.
Later
Eric