The adverb of 'unhurried': doing something slowly and calmly, without rushing. It tells you HOW an action is done — he walked unhurriedly, she unhurriedly folded the letter, they moved unhurriedly through the market.

As with the adjective, the calm matters more than the speed. 'She answered unhurriedly' does not just mean she took a long time; it means she was not flustered, she was in control, she let the silence sit before speaking. Writers reach for it to show a character who is confident, or a place where life is gentle.

Because it is long and formal, it sits mainly in written descriptions, especially in novels. Ordinary speech uses 'slowly', 'calmly', 'in no rush', 'taking her time'. In a conversation, 'he unhurriedly drank his tea' would sound like you were reading aloud from a book.

Position is flexible: after the verb (he ate unhurriedly), between subject and verb (he unhurriedly ate), or at the end of the clause. After the verb is the safest and most natural.