Busy? Not busy? Avoiding engagements (5-4)
I believe the answer is:
fancy-free
'avoiding engagements' is the definition.
(avoiding emotional engagements)
'busy? not busy?' is the wordplay.
'busy?' becomes 'fancy' ('fancy' can be similar in meaning to 'busy').
'not busy?' becomes 'free' (I've seen this before).
'fancy'+'free'='FANCY-FREE'
(Other definitions for fancy-free that I've seen before include "Having no commitments" , "With no emotional ties" , "Lacking responsibility - not currently in love" .)