Very tired to start with, one usually sloshed (3)
I believe the answer is:
sot
'one usually sloshed' is the definition.
(term for a drunken person)
'very tired to start' is the wordplay.
'very' becomes 'so' (I've seen this in other clues).
'to start' indicates taking the first letters.
The first letter of 'tired' is 't'.
'so'+'t'='SOT'
'with' acts as a link.
(Other definitions for sot that I've seen before include "Lush" , "One drinks" , "Habitual drunkard" , "Habitual drinker" , "One's often intoxicated" .)