Hm... you could try doing a search for
shounen-ai + slice-of-life too, since as mentioned most things in the LGBT character in a non-BL/GL manga category have those characters only as a side character.
Some suggestions...
Mainichi Seiten! has some of this, I think, though the first story itself is nothing special, the stories get progressively sweeter as you go on, so you should give it a try if you haven't already.
Otona no Mondai actually has the son of a gay man as the main character, but it deals a lot with what you're looking for, I think. It's also just a great, funny manga in general.
You could also try
Golden Days where one of the major side characters is gay. It deals a lot more with broad life problems though, not just problems exclusive to those in the LGBTQ+ community.
And, yeah, definitely seconding/thirding
Welcome to Room #305! It's definitely the closest thing to what you're looking for that I can think of. It also doesn't just deal with the problems of Hom, the gay roommate, but also deals with the problems of Yoona (a closet lesbian), and some trans* issues as well.