Serial Experiments Lain. I can't quite remember if it is indeed never graphic at all but I think it is a fitting choice.

Madoka Magica is an obvious one.

I think the crown here might go to Takopi's Original Sin. There's hardly any blood, let alone gore. I still never want to watch it again.

Space Battleship Yamato 2199 fits this theme pretty well. The Earth is all but destroyed and humanity only has a small hope. Bonus points because it has an epic theme song and is also fantastic

Kotaro lives alone

I watched it thinking it would be a funny slice of life, which to some extent is is.

But there are also a lot of dark themes around life. Some are obvious at first glance such as the woman working as a hostess suffering from abuse. But there are more subtle ones like Kotaro buying specific tissues (after looking it up later it seems in certain situations children and adults can resort to eating tissues to survive / cope).

Grave of the Fireflies.

Dark meaning depressing.

Ergo Proxy, can't say I liked the story and pace but fitted the dark theme pretty well imo, specially it's artwork.

https://www.youtube.com/watch?v=MrOp6kVX4_w

its dark because Fry under estimates Seymour's loyalty.

Kaya-chan wa Kowakunai It does get dark and it's not graphic.

I was also thinking Dark Gathering, but I don't remember if it's graphic or not

Assassination Classroom is pretty good.

Steins Gate, a romance comedy with psychological helplessness and depression.