Shabby, tense and irritable, doffing cap (5)
I believe the answer is:
tatty
'shabby' is the definition.
(synonyms)
'tense and irritable doffing cap' is the wordplay.
'tense' becomes 't' (abbreviation used in many dictionaries).
'and' means one lot of letters go next to another.
'irritable' becomes 'ratty' (I've seen this before).
'doffing cap' means to remove the first letter (the word's head or 'cap' is removed).
'ratty' with its first letter taken away is 'atty'.
't'+'atty'='TATTY'
(Other definitions for tatty that I've seen before include "of poor quality" , "dishevelled" , "Sounds like King Edward in Scotland" , "Tawdry" , "In poor condition" .)