There is an automatic pressure switch on your original PAR belt driven pump which should turn it off when you have no water running at the faucet. However, on a 40+ year old pump and water system who knows what has happened. The switch could be bad, the intake could be leaking air so the pump will not build up pressure, the internal valves could be leaking and other things.
If it were mine, I would start with a replacement pump. They are under $100 and any parts (when they are even available) for the old PAR would cost you that much. A new strainer would also be appropriate for another $20 or so. They often crack and leak. Put it together with new hoses ($5 or so) and it will probably work fine.
If you can't do the work yourself, any RV repair place can do it for you. Pumps and water systems are universal, not Airstream specific.