Are you sure the tank is full? There is a strong possibility that the sensors are covered in toilet paper. When the paper is dry the sensors say it is empty. When you add some water, the paper wicks up the water and completes the circuit indicating a full tank. This is a fairly common issue. Not sure how your particular sewer systems is setup. If you can, flush the toilet and look down into the tank, and see how full it is. If it looks fairly empty and the sensor says it is full, your sensors are dirty. Another trick, after you dump look at the tank gauge it most likely will say it is full.
If it is a sensor issue, there is some maintenance that might help. After dumping and before going home add 4-5 gallons of water and a minimum of 10 pounds of ice cubes, and drive. I would also add PineSol (2-4 oz) and some liquid fabric softener into the tank. The ice will bounce into the tank sides and help break paper loose from the tank walls. The fabric softener acts as a wetting agent and will break the water tension so the water will absorb and adhere to the items in the tank including the tank walls. The PineSol works as a soap.
You can also purchase a variety of tank scrubbers. One of then you connect to your garden hose and place it down your toilet drain turn the water on and “power scrub” the tank.
Question, what do you do for regular black tank maintenance? After you dump, never let the tank stay dry (except during winter storage in cold climates). My tank is about the same size, I add about 2-3 gallons of water after dumping, add PineSol at minimum; in your case I would also add also add liquid fabric softener, as noted above. While driving home, the water splashes around and pulls the TP down to the bottom of the tank.