Change to F6 in korea

I am currently in Korea and i will regester my marriage here in korea, and in my home country. My question is: If I want to change my visa from C-3 to F-6, I know I have to return to my home country. However, can I go to Japan or China to change it instead? Or are there only specific countries where this can be done?

Also, if I go to my home country, does the process of changing the visa take a long time?