Is Adult being taken here as an adjective here so Nominative? Is "I am AN adult" "Jestem dorosłym" using Instrumental?
Related to „rosnąć” which means 'to grow'. It's just that „rosnąć” has alternative form(„róść”, so „doróść”) and for some reason or another the adjective took from that form(the not used adjective from „dorosnąć” would be „dorosnący”, I guess, which sounds super odd to me – please never use it ;) ).
I typed "I am an adult man" because the prior example was this exact phrase in English. I understand perfectly that this translates to I am an adult and the ending is masculine but in terms of the exercise you may want to add "I am an adult man" to be consistent with the exercises.
When a female voice says "Jestem dorosły" it feels kind of wrong. Well, there are some situations where this might be possible, e.g. when an young woman dreams she is her own grandfather (instead of herself) - but that would be a low probability context. (Or not?)
We don't have the power to make a specific voice say a specific sentence. It's best to think of every audio as someone just reading a sentence. After all if you read a sentence out loud it doesn't matter if it applies to you (even grammatically) or not.
Obviously a woman would say "Jestem dorosła".