Ironically, the one time he's in a relatively good mood and tries to cheer Asuka up, this ends up being exactly the last thing Asuka needs, as her confidence and sense of superiority has been broken and having the unconfident Shinji trying to cheer her up pushes her to the brink.
Then there's Rei Ayanami, whose very existence is an insult against the Laws of Nature and all that is right and good in the world. She can be accurately described as Gloominess Personified.
Sugata from Star Driver definitely fits this, going as far as inadvertently hurting Wako with with his downer behavior. However, considering that he can't leave Southern Cross Island due to being the bearer of Samekh (if he does, his Ninja Maids will be forced to kill him), is also the bearer of a power that can hurt or kill others and put him into a coma, and has to live in fear of Wako's seal being broken and/or her being harmed, this is pretty much justified.
Haruhi Suzumiya: Haruhi's inner workings appear to be quite this at or before the beginning and episode 6 particularly. Being fed up with the boring world, just wishing for something interesting to happen when it just doesn't seem to happen, despite her efforts. In the end (or rather in the middle), she's so frustrated she subconsciously begins to destroy the universe and create a new one. Of course, most of the time she's a genuine Genki Girl.
Ryoga Hibiki of Ranma ˝, who legitimately has a lot to be sad about, is nevertheless prone to such heavy depression he's able to use it to fuel a ki attack - one so powerful it leaves a massive, smoking crater and frequently demolishes his opponents in one go. Yes, that's right, a literal Angst Nuke in the form of a Kame Hame Hadoken.
Natsumi Bekki, the 40 year old teacher for the main characters in No-Rin, acts like a Genki Girl sometimes. However, she will also bring up very private information about her personal life to them, such as being a Christmas Cake, and it often brings down the mood of the entire class.
Hami acts much this way in Dog & Scissors. She Apologizes a Lot, and often talks about killing herself for being alive and being an inconvenience to others.