Is it normal to fill up the grey water tank this fast?

Is the fresh water tank really running dry? The next time your pump won't deliver water, open the fresh water tank drain and see how much comes out. If its several+ gallons, your pickup tube isn't in the bottom of the tank. A quick solution to that is to run a new pickup tube from the pump inlet to a T on the short stubby tank drain hose, between the tank bottom and the tank drain valve. You'll get every bit of water in that tank that way.