This is a double definition - the clue contains two separate definitions that both mean the same answer word. Look for two different phrases that could each independently define the solution.
The clue has two parts that each define the answer: 'Refusal to choose' gives you one meaning, and 'teetotaller's way out' gives you another meaning of the same word. Both phrases point to the same 10-letter word but from different angles.
Answer: ABSTENTION | Double definition: 'Refusal to choose' = ABSTENTION (not voting), and 'teetotaller's way out' = ABSTENTION (avoiding alcohol)