Hi,
You describe a classic RV issue with the in-tank probe systems and how the camper owners have kept their tanks clean "enough". Let's talk about the black tank first. There are metal probes on the side of the black tank that stick inside the tank and touch the liquids, etc. You can see these probes under the camper, as small wires are attached. As crud and paper build up over time on the side walls of the tank, the crud and paper create a path for the tiny electric signal to get connected, tricking the system. It does it off by itself, so the signal is still made when the tank drains, and it will register whichever probe was covered. The black tank is the worst offender of this issue.
The gray tank has the same problem, but the curd comes from long-term use of soap buildup. Yes, the gray tank can get very nasty from scum inside.
You inherited this issue from a prior owner. It takes time for this to happen, and it happens quicker depending on how fussy or not the owner is about keeping their tanks flushed clean after each dumping.
So that is how the problem is created. Now, what do you do about it?
1. Some folks ignore the black and gray as they know the problem. Everyone looks at this differently.
2. You can use a tank cleaner product to eliminate the crud in each tank. You fill the tank full of water with the cleaner and wait; overnight or longer is at least required. This is not a fast process. I have used the Happy Camper Extreme Tank cleaner product, and it works for my friend's camper I used it on.
HAPPY CAMPERS RV Holding Tank Treatment. It is sold directly from them and elsewhere; Amazon may also have it. There are other cleaners also, Tank Techs being one of them,
Just Science | Innovations that help, not harm
I would not suggest using the acids, like Drano or other drain cleaners; this is a different issue. You want to use the methods of digesting enzymes/bacteria.
As stop-gap measures, you can see the black tank inside the camper, turn the water pump or city water off, use a flashlight, push the foot pedal down, and look down the hole to see how full the tank is.
You can't see the gray tank, but it will back up into the shower basin when full and not drain on most floor plans, as the shower is the lowest point. There is an exception: the T264SR and possibly the T260SR may back up into the kitchen kink in place of the shower. The pipe routing is different on that floor plan, especially regarding how the roof tank vents into the drain piping.
I can type some more later, out of time right now, about how to prevent the issue from happening in the future once you get the tanks cleaned out. It is all about how you flush the tank and what you consider "good enough".
Hope this helps
John