Ask yourself whether what you want is really possible. They just might introduce you to some cute Korean guys.
They both dress me up as I like to be in workout clothes more. You are dating in Korea, so naturally you will be expected to do things the Korean way.
Your relationship together should not be based on the fact that you already speak semi-fluent Japanese. Don't Correct their English unless they ask you to. The mistakes people make in Culture Shock relationships is that they make a big deal out of the differences, and this plants the idea that you really are too different to be together.
Yes, they might still be learning, but you don't need to be constantly correcting their English unless they want that extra help. And it is not the culture that is at stake here, it is your obsession with the different cultural differences and treating them differently because of it that will drive you two apart. Don't watch TV shows like "Fresh Off The Boat" together.
So if you want to have a serious relationship with someone who is Asian (or African, or Middle Eastern, or Kryptonian - honestly, these tips will help any couple with a case of Culture Shock Love). Meet their friends first, before you meet their family members. Find the things you have in common - sports you both do, activities you both, a love of art / art galleries, and build upon that connection.