Old musician woozy, measure of drink found slightly lacking (5,9)

I believe the answer is:

dizzy gillespie

'old musician' is the definition.
(I know that Dizzy Gillespie was a instrumentalist)

'woozy measure of drink found slightly lacking' is the wordplay.
'woozy' becomes 'dizzy' (synonyms).
'measure of drink' becomes 'gill' (gill is a unit of liquid volume).
'found' becomes 'espied' (espy can mean to spot or find).
'slightly lacking' means to remove the last letter.
'espied' with its last letter taken away is 'espie'.
'dizzy'+'gill'+'espie'='DIZZY GILLESPIE'

I've seen this clue in The Times.

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