Moaned, beginning to get annoyed about signs of doziness (8)
I believe the answer is:
grizzled
'moaned' is the definition.
(grizzle can mean to moan or grumble)
'beginning to get annoyed about signs of doziness' is the wordplay.
'beginning to' indicates taking the first letters.
'annoyed' becomes 'riled' ('rile' can be a synonym of 'annoy').
'about' is an insertion indicator.
'signs of doziness' becomes 'zz' (zzzzz can represent snoring).
The initial letter of 'get' is 'g'.
'riled' enclosing 'zz' is 'rizzled'.
'g'+'rizzled'='GRIZZLED'
(Another definition for grizzled that I've seen is " Streaked with grey".)