Welcome aboard L&R, congrats on the purchase.
Agreed, have never had a problem with mine running dry - the pumps should self prime. When switched on the pump builds pressure in the system, and if no valves are opened, the pump should stop running - until a valve is opened. Does the pump ever stop running while still switched on?
Nevertheless, if the tank has water, and the pump is running, and there are no leaks - can only be a couple of things. The pump likely has a filter on it, probably a clear plastic screw on/off cover with a fine screen within - make sure that screen isn't clogged with spooge. If that is clear then possible each fawcet is clogged - though not likely. Also try filling the holding tank to the top. If it is a pump prime issue perhaps the extra weight of the additional gallons will force water into the pump. If all that fails - new pump.