Of the recommendations given to you, I will second the following: Mushishi, Planetes, Eve no Jikan. And with the disclaimer that it's more of a romantic comedy than a mature drama, Nodame Cantabile is quite good. Seirei no Moribito is also good but somehow I feel the setting/plot isn't really what you're looking for, but maybe I'm wrong.
I will also throw in my own suggestion as well: Millenium Actress. It is not a series, it is an anime movie, but if you want a deeply rewarding drama about adults, it's one of the best.
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