Poking around with my multimeter, I discovered the following:
The voltage at either motor terminal is small and random (floating) when the battery is connected - this might be noise with my DMM or just some other interference but it's not zero.
When switched on (and it always switches on in auto mode), the voltage (measured between the black lead on the power supply plug and the motor terminal) is 6V. On both. It may be slightly variable which would explain the slight turning of the motor. Still trying to get my head around the H-bridge to see what is happening.
It would appear from here:
that the inputs each side are going low thus driving each side of the motor to high, whereas it should be one or the other. Not quite sure how this changes things with regards to the tilt test.
In manual mode, I managed to clip the DMM negative lead to the black lead input so I could press the buttons for up and down. Each drain is going to 6V with a press of the button (on both MOSFETs for any given motor) and back to zero once released. I'm wondering if the MOSFETs were ever at fault and the issue lies elsewhere...