Elfen Lied. Bebop.
Ghost in the Shell:SAC is sorta drama, I guess. Kinda hard to define "drama". You could even consider TTGL drama if you wanted to.
Shigofumi, Tales of Memories/Melodies, Welcome to the NHK were some of the most emotional animes for me. Shinigami no Ballad wasnt bad either if you enjoy Shigofumi.
None of them had great production value, and I can imagine certain people not liking them at all.
Finished watching Gankutsuou, it's more on the gay side of the spectrum, but it has some pretty good drama.
It's based off the count of monte cristo, though the artstyle isn't really easy in the eye, and like I mentioned earlier, gay.
Haydee <3 though, can't get enough of docile doll like girls~
On the contrary, I would highly recommend Gankutsuou. The art is unique. In fact I can't recall an anime in recent history that has as much style and sophistication as this. As for drama... The count is easily one of the best drama characters in literature. From early on, it's apparent how twisted he is, yet you can't help but want him to fully execute his revenge scheme.
not sure if these fall under what you're looking for, but would recommend :
Samurai-X: Trust and Betrayal
Air
Mushi-Shi
Welcome to the NHK
I also second that recommendation to watch Welcome to the NHK. Excellent anime right there. :D
I hated reading the manga just because how close it hits the first time I read it :/
The anime diluted a good bit of the content from the manga too.
I haven't seen the anime, but I've heard it follows the manga pretty closely, so Monster is prolly a good bet.
Welcome to the NHK does come to mind as a drama too. Witch Hunter Robin maybe. Ergo Proxy n' Wolf's Rain maybe, depending on taste. Both of those, to me, had some good episodes and all, but couldn't seem to wrap things up too well.
Most anything Ghost in the Shell, or if you're looking at movies, most anything Oshii has done.
Similarly, Evangelion could be good if you're one of the two anime fans that haven't watched it yet. Part of the reason people have such a split opinion over it is -because- it's not just giant robots duking it out.
Kino's Journey, Haibane Renmei, Serial Experiments Lain also come up as I mentally sift through my collection, lol. Lain particularly might be one to look into if you haven't watched it (it is an older one).
Alot of the obvious ones have been said, but i recently watched Gun X Sword cause it was a streamed anime on Netflix and I really enjoyed it, very Trigun like but was a good story and was fun to watch.