I found and fixed the problem. For anyone in the future doing a search on Perfect 360 - P360 - joystick problems, here is the solution:
I took out the 4 silver screws on the bottom that hold the electronic assembly to the base. I inspected the 8 sensors on the PCB and found that one sensor was bent back so that it was facing up instead of out towards the shaft. I simply bent the sensor back to where it was facing the shaft, and voila! instantly working P360 joystick.
These are nice joysticks.