I then gently stuck a broom handle down the hole and met a solid surface.
I don't know how your RV is configured, but in mine, the toilet is positioned directly above the black tank, so when you open the toilet flapper valve and look down when the tank is empty, you can see the translucent plastic bottom of the tank.
So if I suck a broom handle down my toilet hole, it also would strike a solid surface - the bottom of the tank.
If your tank is full of water, you have to potential problems:
1) The tank is not draining. The way to verify this is as was said above: Simply open your drain valve with no hose hooked up, have someone flush the toilet. Water should come out of the waste gate. If no water comes out, you have a blockage.
2) You have a water leak somewhere that is filling your black water tank. In some RVs I am given to understand that not just the toilet dumps into the black tank. Sometimes the bathroom sink or other things can also dump into the black tank. If this is the case on your RV, and you have a leaky faucet, it could be filling up your black tank.
Steve