It appears that you have a pretty 'crappy' situation to deal with now.
Chances are that you have built up one heck of a dam at the drain opening that is preventing the tank from emptying.
There are professionals that have specialized equipment to clean out the clog and the tank. I would hire them to come to your RV and do the deed.
Then with what you have learned here, with the proper usage and water, etc. you should be able to keep your tank flowing the way it should.
It is always better to keep both tank valves closed unit the tank is full or 3/4 full. Then dump the black tank first and when that is empty leave the valve open and then pull the grey tank valve to empty that tank. Once both tanks are empty, close both valves for the next time.
Using a clear fitting at the valve before the hose is a great idea, it gives you a way to monitor the action going on.
Using Calgon water softener helps too along with using LOTS of water.
You may have to dump the tanks more frequently but water is your friend.
Dr4Film ----- Richard