Unless i had missed it, i haven't seen
Hanbun no Tsuki ga Noboru Sora mentioned yet in this thread. It might be an ultimately bittersweet show, but the short 6 episodes it spans are very much worth it. A touching (and sometimes heartwarming) romance/drama in hospital setting. This remains as one of my favorite shows up to this day.
I would also second the aforementioned
Iriya no Sora, UFO no Natsu. Just like the above title this one spans only 6 episodes, but it manages to tell a touching story of two people facing the insurmountable odds of fate their specific situation brings forth. It is very much a drama title, but well worth watching.
Fruits Basket is something i believe you would find enjoyable. The series does contain less actual romance than you are probably looking for, but the implications and relationships are there, along with an interesting set of characters and plot to string things along.
Another title i have yet to see mentioned is
Gift ~Eternal Rainbow~. (Don't believe the 'harem' label on it - whomever labeled the series must not have actually watched it and just used the admittedly misleading image as a reference). Basically a love triangle (but not quite) comedy/drama with a wee bit touch of fantasy to the setting. Features what i consider to be one of the more likable male leads out there, who also happens to know which of the two girls he likes from the start and actually acts upon it, which is something we get to see once in a blue moon
I see
Sola on your 'to-watch' list as well, which gets my nod of approval - the series got a bit weird towards the end, but it was an ultimately fulfilling tale of romance and drama (mixed up with some fantasy elements) and featured some quite enjoyable interactions between the male and female leads.
ef - a tale of memories is a good choice as well. The story tells two separate tales of relationships at once, and the highly stylized and (sometimes overly) artistic visualization of the show might take some getting used to, but chances are you would like it by the end
(there is a currently ongoing second season
ef - a tale of melodies, but, in my opinion, it falls somewhat short of being as highly acclaimed as the first season was ... you may still end up liking it, so consider it if you end up watching and liking the first one).