So... is Oooku finished or not?
NO, it is not finished.
Why?
The anime finished after the manga:
Anime finished: June 29, 2023
Manga finished: December 28, 2020
... But MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Oooku presents an alternate Edo-period Japan where a mysterious disease decimates the male population, leading to a matriarchal society.
The current shogun, Yoshimune, is curious about the origins of this inverted world and begins to investigate historical records.
Her investigation uncovers the story of Iemitsu, the first female shogun, and the circumstances that led to the unique traditions and power structures in place.
The anime explores the history of the Oooku, the shogun's inner chambers filled with thousands of men, and how it came to be under a female shogun's rule.
A central figure in this historical shift appears to be Arikoto Madenokoji, a young monk whose life becomes intertwined with the shogun and the Oooku.