I was going to recommend 1 Litre of Tears, but I see you've already watched that one.
I have a hard time saying what's a classic or not, especially since I'm very new to dramaland. But here's some of my (random) recommendations, hopefully you won't be too disappointed if they're not really classics, and movies.
Orange Days (Japanese drama)
Byakuyako (Japanese drama, I haven't seen it yet though)
HERO (Japanese drama, I started this but I got too busy to continue)
A Millionaire's First Love (Korean movie)
Yuuki (Japanese movie)
and I'm going to throw in Nodame Cantabile in the mix too. I'm being really biased here to be honest, because I don't think it would be called a classic.
I'm sorry if this isn't what you are looking for.
