Hehe, I'm not a native English speaker, but that "human" part feels a bit odd to me too – "I am an adult.", should be enough, I feel… But maybe I'm wrong. ;-)
I used "person", that's how I usually translate "człowiek". And "mężczyzna" for "man".
Another correct answer should read - I am an adult. In English we do not say 'I am an adult person'
Can a woman say this sentence? If not, how would she say it in the feminine case?
"Jestem dorosłym człowiekiem" - yeah, sure. A man is a human and a woman is a human. But somehow I think most women would rather say "Jestem dorosłą kobietą" = I am a grown/adult woman.
As a Polish woman I would say:
"jestem dorosłym człowkiekiem"- only when I really want to make a point of being human
"jestem dorosłą kobietą" - when my womanhood is important
"jestem dorosła" - "I am an adult"
"jestem dorosłą osobą" - I am an adult person
two last ones seem to me like ones I would use without giving attention to either my equal right as a human being, or my gender.
That sounds completely like a quote from "Star Wars", but... that's not a reason to reject it, I guess. Added.