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!
Disco Elysium shattered me half a dozen times. Two moments in particular made me step away from my desk and go on a walk. It's the best-written game I've ever played.
Firewatch. Gorgeous, emotionally honest game that wrings you out.
Inside. More of a stressful, cathartic cry, but it applies.
Mass Effect 3 - but only if you play the first two. Incredible story that beats you over the head with yearning .