The two answers to this flat are compound words in which the component words are opposites. That is, the first part of the first compound word is the opposite of the first part of the second compound word. The same is true for the second parts.

I guess before I _ _ _ _ _ _ my operation,
I'll have to _ _ _ _ _ _ _ _ my qualms about sedation.

Answers:

UNDERGO
OVERCOME

Credit:
The Enigma
NPL Magazine - Oct. 2012
ALEX