Basically what the title says. I've only played two games like this so far (Silent Hill 2 and OMORI) and am looking for more.
Could be games, ROM hacks, mods of existing games, etc.
EDIT: So many great suggestions! Getting a bit overwhelmed by how many there are, so I'm going to need to find a way to put this in some kind of list with an order to play the games in. By all means, keep 'em coming, though!
came here to say spirit farer. that game is responsible for normalizing me hugging my spouse for no reason besides it feels good. Also definitely made me cry more than pretty much any other work of fiction in any medium.