After some tinkering and realizing I needed to learn more about transistors, I figured out how to drive these coaxial switches that are available for cheap on eBay (thanks for the tip @k5em !) using 3.3v logic levels. tl;dr I needed a 2-stage high-side driver for each side instead of a single low-side transistor. The "click" sound is the switch changing positions and the green LED shows the actual pulse that is sent (100ms).