And on mine I have two ways to drain the rear grey tank. There is a valve in the front so I can drain all three tanks from the one drain outlet. Or, I can drain the rear tank from it's own pipe at the rear of the trailer. Confused yet? Yep, I was at first too.
Yes, the sensors can be wrong. How old is your rig? If several years old then the sensors may not be working. Look around for methods to 'clean' the tanks so the sensors will work again. Our old RV's were 'old'. Sensors didn't really work at all. I'd empty the grey tank when it backed up into the shower
Peek into the toilet to see how full the black was