How can I switch this audio jack, using its own mechanical switches, without creating feedback?
You don't even need a semiconductor.



In this circuit, when no jack is connected, GPIO will be pulled down to (almost) ground R4. When you insert the jack, R4 becomes disconnected, and R2 weakly pulls GPIO up to AVDD.

In your case you'd have VCC instead of AVDD. And you'd have AUX-ON instead of GPIO.