Full of sadness and angst:
Autumn in My Heart
Stairway to Heaven
Tree of Heaven
Jang Ok Jung
That Winter, the Wind Blows

I'd suggest currently on-going drama: My Love, Eun Dong. They've only released six episodes and I've watched the first episode. Eventhough I haven't watch the remaining episodes, I can tell it's going to be pretty sad. Judging from the comments and reaction of the drama, too.

I honestly don't watch a lot of K-dramas, but I heard that Winter Sonata was pretty sad. Like I said, I never watched it (I've only listened to the OST), but from what I heard, it was sad.