Thanks u/marklein for helping me identify a starting point to measure Q1 which is a MOSFET. It's marked with XORB, and searching "XORB MOSFET" led me to Digi-Key's website for this product, and page 1 of the datasheet indicates that the bottom 2 terminals are Source/Drain.

This article explains a functioning MOSFET in the "off" state will have no electrical path between Source/Drain. This youtube tutorial explains how to conduct the measurement, which showed a voltage reading between the terminals, meaning the MOSFET is blown. These are only $0.43 at Digi-Key, but the product reviews for the neon sign show this is a widespread issue, so it's likely a poor board design and replacing the component will just lead to the same issue later on (plus I don't own a soldering iron yet).

I'm just going to buy a 12VDC LED dimmer switch with an LED strip splitter (I'll use a 2pin because I only need to power 2 sets of lights, though 4pin seems more common). This article from LED Supply helped me understand a bit more about the dimmer switch and splitter.

~$25 of parts for a fast solution, can definitely be done cheaper but not worth since I believe the cable / PCB design is faulty to begin with. But it beats replacing the entire assembly for ~$125