Japanese:
Hana Kimi- comedy romance - for me it's a classic
First Kiss- I really liked the lead of the romance story, such a nice, responsible guy.
Nodame Cantabile- for me the best drama ever!
I second Love Shuffle- it's something different from most dramas.
Proposal Daisakusen- I see you've already started, for this one you'll need some patience but it's worth it.
Korean:
Iljimae- great drama, but it left me emotionally unstable..
Ojakgyo Brothers- I know it's super long, but there's so many different stories in it, that it feels like watching several dramas at once. Romance was very well done.
City Hunter - I don't see it on your list, and that's a must-watch! Both for romance and for action.