Bands I'd struggle with, but their are plenty of individual artists that sing about such things. You should check out the LGBT songs section to see if there's anything you like. I mean there is "A great big world" who do have some LGBT related songs, but they're a duo rather than a band.
Two gay bands I can think of (both from some years ago) -Pansy Division - sort of a punk/hard rock feel -The Fabulous Flirtations (not to be confused with the 50s group). Amazing acapella group with lots of fun and funny songs. -Sweet Honey in the Rock - also acapella, though more folky. Several lesbian members, most of their songs are about social consciousness topics.
Hunx and his Punx is a band that I might suggest, the irrepressibles have a album called Nude and it's got a song called Two Men in Love and the album is baroque pop. So definitely melancholy.
For more melancholy, I'd check out The Smiths. I'm not sure if the lead singer Morrissey ever said if he was gay or bi, but lyrically, they cover the ground better than any band I can think of.
Judas Priest's Rob Halford is gay and they sang about several gay themes back in the 70s. Check out the songs Raw Deal and Jawbreaker.