I'm looking for a specific type of ambiance in my manga that I might describe as dark and melancholic at the same time. Perhaps the closest word that might fully capture what I'm looking for is saudade. I've linked it for reference.

Some of the manga that I felt were particularly close to this specific atmosphere were:
xxxHoLic
Nobody Knows (LEE Hyeon-Sook)
Pet Shop of Horrors
Clover (Clamp)
Hanamatsuri

I know this is a very difficult request, so I really appreciate any help that you could provide!

I am a piano
Kuro Bara Alice
The Tarot Cafe
Flowers of Evil
Ludwig Kakumei
Antique Romance

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As they played the game...I waited for someone to call "onigiri." But no one called. I was very little then. I had almost forgotten about that. [...] But they knew...there would never be room for an onigiri...in a fruits basket.
-Tohru Honda, Fruits Basket

Hitsuji no Uta?

Yokohama Kaidashi Kikou - not dark at all, yet has a very nostalgic feeling of loss... a world at the end of an age
Natsume Yuujinchou - haven't read the manga myself, but I'd assume it's at least as good as the anime

I second Ykk, Hitsuji no Uta and Flowers of Evil. I also add Kaiba anime which is extremely good and Hotel.
Also, I premise that this is probably very subjective, but I'd consider also Pluto and Vampire Juujikai.

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What I like in Trivial Pursuit style. Pick your category:

Well not really in the beginning, but towards the middle i think Kare Kano is pretty dark and sad. If that is to say what you're looking for.

Definitely try reading Watashitachi no Shiawase na Jikan.
Also, Annarasumanara, Hoshi no Koe, Vampire Juujikai.

A, A'
Poe no Ichizoku
Oyasumi Punpun
A Million Pound Love
Twin Spica

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