Miser getting drunk initially wants another drink (8)
I believe the answer is:
tightwad
'miser' is the definition.
(I know that tightwad is a type of miser)
'drunk initially wants another drink' is the wordplay.
'drunk' becomes 'tight' (slang term).
'initially' indicates taking the first letters.
The initial letters of 'wants another drink' is 'wad'.
'tight'+'wad'='TIGHTWAD'
'getting' acts as a link.
(Other definitions for tightwad that I've seen before include "Cheapskate" , "Skinflint" , "Mean person (US)" , "Miserly person" , "Miser (sl.)" .)