Move unsteadily after beginning to drink more rum (6)
I believe the answer is:
'move unsteadily' is the definition.
(I've seen this before)
'after beginning to drink more rum' is the wordplay.
'after' says to put letters next to each other.
'beginning to' says to take the initial letters.
'more rum' becomes 'odder' (rum can mean odd).
The initial letter of 'drink' is 'd'.
'd'+'odder' is 'DODDER'.
(Other definitions for dodder that I've seen before include "Walk unsteadily along the Dublin river?" , "Progress unsteadily (due to age?)" , "Move in frail manner" , "go falteringly" , "move unsteadily" .)