Go off theologian overcome by drink (5)
I believe the answer is:
addle
'go off' is the definition.
(eg an egg might addle)
'theologian overcome by drink' is the wordplay.
'theologian' becomes 'dd' (Doctor of Divinity degree).
'overcome by' indicates putting letters inside.
'drink' becomes 'ale' (I've seen this before).
'dd' placed into 'ale' is 'ADDLE'.
(Other definitions for addle that I've seen before include "Confuse; become rotten" , "Confuse - decay" , "Make confused - go bad" , "(Of eggs) turn bad" , "(Of egg) become rotten" .)