Water pump continues to run on

I had a similar issue... Replaced water pump, check valve for city water, and tightened water connections and problem still existed. Turned out to be the fresh water pick-up hose from the from the holding tank to the water pump. The fitting on the pump was tight but the clamp that held the hose to the fitting was a little loose... It was letting a little air get into the line causing an air pocket in the pump. I tightened the clamp and the problem was fixed. Check the hose fittings... If tight, you have a leak somewhere between you fresh water tank and output of your water pump. By the way, when I was hooked up to city water I had no problems with loosing pressure, only when I ran off the pump.