Proposal Daisakusen and its Korean version, Operation Proposal had nothing to do with reincarnation, it was only time travel. The guy didn't die to be reborn, but was given the chance to go back in time to make things right with his beloved - same story as Manhole, only better made. A 19-year old Yoo Seung Ho was impressive in his first leading adult role.
I've watched more than 150 Korean dramas so far and, although time travel is very popular, ghosts who visit the living are very popular, but it's strange how the reincarnation theme hasn't been exploited more by writers!
Faith/The Great Doctor and Queen In Hyun's Man was also NOT reincarnation but time travel. (I haven't watched Dr. Jin but I think that's the same thing).
Rooftop Prince blends time travel with a little bit of reincarnation, because the guy from the past travels into the future, but there is his reincarnated self in coma all this time (and that's the reason for the time travel of the ancient version of the soul)
In Alice: The Boy from Wonderland the main lead is a ghost (the story is extremely complicated, you have to watch it twice and read countless comments to understand what's going on), but he gets reincarnated at the end (that blurry thing is a spoiler, so if you don't mind, click on it to read it).
Gu Family Book also has reincarnation only at the very end, and everyone was pissed off at the sloppiness.
THESE ARE THE TWO I CAN SUGGEST
Goblin had reincarnation of two women. One is made to remember the past from some point onwards, the other will be reborn in the future and we have a glimpse of her reincarnated self.
Chicago Typewriter had three reincarnated people and one ghost from the past.
From all of the dramas I watched, these two where reincarnation was a major theme.