Check valve is in the water pump. When it fails, the fresh water tank will fill when on city water. The problem might be in your Anderson Valve. The black tank flush has an air gap valve. Pressure does remain in the hose leading to it. I had a one to two hose connection "Y" for the black tank flush. I would release the pressure using the second side as to not get wet by disconnecting the hose with pressure on it.
Not an issue filling the fresh tank. It's the pump pushing water out of the water hose inlet.
Yes, the check valve in the water inlet is either stuck, or broken.
You might give Anderson Brass a call to see if they'll send you a replacement check valve to install - maybe for free.
Or you could screw in a check valve like
this one from Amazon. Is says brass plated, so you might not have to worry about lead content. If that's a concern, you can no doubt find a lead-free garden hose check valve for a few more $.
On the black tank, you should make sure that water is actually getting to the black tank. Watch it come out a clear sewer outlet adapter. You might have a kinked hose, perhaps near where it enters the tank.
If no water goes through the flush, the vacuum breaker might be hooked up backwards. But check water flow before going down that path.
