Hi Jan and Bill,
I think the normal behavior of the gray #1 tank is that when full, the shower will have water on the floor, and it will not drain. If the drain pipe is cracked or fits poorly going into the gray tank, that would explain both the slow draining and the water in the underbelly.
The black tank issues are likely completely separate. It could be as simple as a little bit of TP trapped in the gate valve, keeping it from closing completely. When that happens, some or all of the water you flush just leaks out leaving mostly solids behind in the tank. That makes it much harder to clean the tank, and more likely to get clumps that clog the outlet at the gate valve. I've had that happen a couple of times and usually have to flush and back flush the black tank 4 or 5 times to get it really clean. There's an add-on that make it much easier to manage.
If you get a clear extension and an external gate valve to put on your sewer outlet, like
this one, you can see what's coming out of the drain to know when it's running clear. It has a fitting to attach a hose so you can run water back toward the gate valve to help clean it out. There's also an external gate valve which will solve several problems. First, if you close the external valve, even with water leaking through the black gate valve, you won't lose all the flushed water and dumping the tank will be easier. Second, when you remove the cap to hook up the sewer hose, you won't have black water come out on your shoes. You just open the valve after connecting the hose. Third, when flushing the tank, the external gate valve will allow you to back flush the tank. Here's how I do it:
In normal operation, dump the black tank and run the internal black tank flush for several minutes until what's coming out is clear and running at a constant, relatively slow rate. Then close the external valve and keep the flush running for 4 minutes or so (depends on how much water flow you have - I estimate I'm putting in 4-5 gal/min if the campground has good pressure, but that's with high volume pressure regulators). Open the valve and drain again to get it running clear. I normally do this 3 or 4 times to make sure everything is out of the tank, before I close the black tank gate valve. Then run the black tank flush for a minute to put a few gallons of water back in the tank.
Sometimes the water coming out the drain will slow down to almost nothing, indicating a clog at the black tank valve. Allow the black tank flush to continue flushing, but keep track of time so you don't overfill the black tank. Usually the clogs will break up within 2 or 3 minutes at most. If you get to 4 or 5 minutes, turn off the internal flush and blast water through the flush pro's fitting (you need a check valve to be sure you don't contaminate the campground water by getting sewage flowing back through the hose).
BE VERY CAREFUL when dumping the tanks. Do not let yourself get distracted by other people or start looking at other things. It's too easy to forget you're filling the black tank.
And if none of this works, you may just have a leaky black tank valve - it happens. There's also a possibility that when the black tank opening was cut, the cutout disk fell in and is up against the gate valve. Next time you're at the dealer, they can open it up and resolve the problem under warranty. In the meantime, that external gate valve will let you manage the situation.