Depending on how your rig is plumbed, the tank drain may be downstream form the pump outlet - mine is that way (water drains faster with the pump on). That would mean that opening the drain allows the pump to pump freely, and closing it creates back pressure and the pump shuts off. That would indicate a pinched water line somewhere between the pump and faucets, or a clogged filter if you have one.
If your tank drain is direct from the tank rather than in the plumbing lines, I wonder if the water tank vent has become plugged somehow? That would eventually cause a vacuum in the tank as water was pumped out and the pump couldn't pull suction against it. Opening the tank drain would break the vacuum and let water flow. Seems like the pump should run for a while and slowly stop, if that were the case, though.
Former owner of 2004 American Tradition and several other RVs
Home is West Palm Beach, FL