Heading off, slightly intoxicated after quiet drink (6)
I believe the answer is:
sherry
'drink' is the definition.
(sherry is a kind of drink)
'heading off slightly intoxicated after quiet' is the wordplay.
'heading off' means to remove the first letter.
'slightly intoxicated' becomes 'merry' (I've seen this before).
'after' means one lot of letters go next to another.
'quiet' becomes 'sh' (both can be said to tell someone to be quiet).
'merry' with its initial letter taken away is 'erry'.
'erry' after 'sh' is 'SHERRY'.
(Other definitions for sherry that I've seen before include "Pre-dinner aperitif as from Jerez" , "Christmas drink?" , "Spanish fortified wine" , "Preprandial drink" , "Alcoholic appetiser associated with Jerez" .)