Busy nurse having good time with daughter (7)

I believe the answer is:

engaged

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'busy' is the definition.
(I know that busy can be written as engaged)

'nurse having good time with daughter' is the wordplay.
'nurse' becomes 'EN' (abbreviation for state-enrolled nurse).
'having' is a charade indicator (letters next to each other).
'good' becomes 'g' (abbreviation).
'time with' becomes 'age' (an age is a period of time. I am not sure about the 'with' bit.).
'daughter' becomes 'd' (genealogical abbreviation).
'en'+'g'+'age'+'d'='ENGAGED'

Can you help me to learn more?

(Other definitions for engaged that I've seen before include "Taken on - betrothed" , "Employed - to be married? (7)" , "Spoken for" , "as a result of a proposal" , "Hired; busy" .)

I've seen this clue in The Guardian.

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