Water pump continues to run on

Are you losing pressure back through the pump diaphragms? This would not move water so it would be quiet but the pump is trying to regain the pressure. There could be a piece of grit, trash or even a tiny hole causing it. there are rebuild kits on most makes

You might not want to hear this but you have a leak somewhere.

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.