Had worried an awful lot about being outmoded (3,3)
I believe the answer is:
old hat
'outmoded' is the definition.
('old-hat' can be a synonym of 'outmoded')
'had worried an awful lot about' is the wordplay.
'worried' indicates anagramming the letters.
'an awful' indicates an anagram.
'about' means one lot of letters goes inside another.
'lot' with letters rearranged gives 'olt'.
'had' with letters rearranged gives 'dha'.
'dha' put inside 'olt' is 'OLD-HAT'.
'being' acts as a link.
(Other definitions for old hat that I've seen before include "Stale - out of date" , "Dated" , "no longer fresh" , "Passe" , "Bowler, possibly" .)