Desperately dry, I clam up in party (5,5)
I believe the answer is:
plaid cymru
'party' is the definition.
(Welsh political party)
'desperately dry i clam up' is the wordplay.
'desperately' indicates an anagram.
'dry'+'i'+'clam'+'up'='dryiclamup'
'dryiclamup' anagrammed gives 'PLAID CYMRU'.
'in' acts as a link.
(Other definitions for plaid cymru that I've seen before include "Welsh political party" , "party in Wales" , "Welsh Nationalist party" , "party in UK" .)