I've met a lot of foreigners in Seoul who work in various fields. I think the easiest way to get a non-teaching job in Korea at least (can't speak for Japan) is to have a degree in the field, a lot of relevant past experience in that field, and to be at least at a conversational level when it comes to Korean. All of which it seems like you're on your way to accomplishing, so I would say you're on the right track. Internships are abundant as far as I know. I've met many people who are doing one or two year internships at big name Korean companies, and the majority of those people did not know Korean beyond how to (badly) say 안녕하세요. One woman I know was interning at what is considered a "dream job" by many Koreans while she was completing her Masters at Korea University, an internship that lead to her becoming a contracted employee upon completion of her degree. So, if you know where to look and know how to sell yourself well, internships shouldn't be a problem in Korea.
If you're looking for a stable, non-contract job (by non-contract, I mean a job where you don't have to sign a new employee contract every year), that will be more difficult as a foreigner. Korea is still a fairly conservative country when it comes to foreigners. Meaning that many companies might not be interested in hiring a foreigner as a full-time employee. A lot of this has to do with their unwillingness to cut through the red tape of sponsoring your visa, but the situation is particularly difficult for those foreigners who are not white. I have a friend who is currently on a D-10 visa, which is a visa that allows foreigners to stay in Korea for a few months as long as they are actively looking for work. She is a black Italian with a masters degree from a Korean university who has relevant work experience and speaks excellent Korean. She recently attended a job fair and I was saddened when she told me how many employers wouldn't even glance at her resume before dismissing her as someone the company would not be interested in. Even more horrifying was her account of another black foreigner with an amazing resume and experience in Hotel Management who has been in Korea for seven years and speaks the language fluently. He was told by a recruiter from a prestigious hotel that they would love to hire him because of his impressive credentials, but because he's black, they were only willing to offer him the position of bellhop.
That's all I've heard about the business world in Korea. I'm just a teacher myself, so all of this is coming secondhand through me. Just thought I'd let you know what I've been told by friends who are doing what you want to do. Hope it's helpful, and good luck!
