Me and my students had this problem for about 4 months. For some reason, the printer would say that the 'center tray is full' even though the center tray was not full at all! I've tried resetting and turning off the printer, but nothing is working!
Have you updated the firmware?
If the firmware has been updated to the linked version, it is safe to say the issue is hardware and that the sensor is problematic.
I don't have a Phaser 7800 to check, so I can't state where the sensor is, it could be a light sensor, or a toggle switch, but it is probably visible and should move freely.
The sensor what are you searching for is part nr.14 Full Stack Sensor (Q77-125) in the Exit 2 assy.
It looks like an A/P finisher. The center tray is the one below the top tray. Something is wrong with the sensors there.
By the way, the exit 2 is on the left side of the printer above the fuser.
Sorry if I am wrong, but I looked at the Jason picture (I zoom it) and I saw that the mimic of the machine show the paper on top of the Orizontal transport.On the other hand I know that the trays on finisher are: Top tray, Stacker tray and Booklet maker tray.
All the best!