Sorry to hear things have been rough for you. I hope they get better soon. Here are the games that made me sad but also made me want to continue to the end to feel the feelings:

That Dragon, Cancer (not on GOG) This was a very strong cathartic experience for me when I played. It's about a family dealing with their baby's cancer. There are a few annoying minigames, but for the most part, man, it hit me hard.

Brothers: A Tale of Two Sons (not on GOG) I see some others have recommended this, and I do too. This is about two boys who go in search for medicine for their father. As they get farther from home, things get more surreal, huge, and overwhelming. It does interesting things with its controls.

Papo e Yo (not on GOG) A boy has a giant monster friend who gets violent when he eats a certain food. This is a beautiful puzzle-platformer with authentic Brazilian street art based on the author's experience of living with an alcoholic father.

Human Resource Machine Gameplay consists of programming, but the theme is of working at a meaningless office job 9-5 with only the hope of retirement to drive you to continue.

Dropsy You play a mentally-deficient clown who has been accused of burning down the circus tent with his family inside. Point-and-click puzzles to try to get people to stop hating you.

Fran Bow You are a young crazy girl dealing with frightening imagery. Point and click to nevertheless go on despite your mental state.

Massive Chalice A turn-based strategy game about leading a centuries-long war against a horde of demons. Your fighters grow old and die and are replaced by the next generation. It plays similar to XCOM but with a more serious tone and also (you may not think this an improvement, but I did) none of the endless cut-scenes interrupting the battles.

Shelter You play a mother badger leading the kits on a journey. You must keep the offspring safe and fed, and you will fail.

Unrest A minimalist RPG. You play a series of characters at key decision points of their lives. The options they have are all pretty bad (for example, the first character is a young girl being forced into an arranged marriage).

Reus In this game, you have three or four elemental giants that can change a planet. You try to get combos of things near each other to spawn improvements. If you are me, the worlds you try to create will get better to a point then spiral out of control and end in death and extinction. But you may actually good at the game, so results vary.

Din's Curse This is a procedurally-generated action RPG where you have been a terrible person, so in your afterlife you are condemned to fight as a champion of the god, Din and try to save a series of towns that are full of poverty, thieves, and monster attacks.