Hmmm..not exactly sure about what a "Shur-Flo" looks like....I have the older style "PAR-Jabsco" dual reservoir water pump on my 69 Airstream.
I know they usually always have a "pressure switch" that kicks the pump on demand when pressure drops...e.g. when you open the spigot.
Thes pumps will cycle anywhere from every few seconds, to every ten minutes, if there is some kind of a compromise in the seal in the water system. Basically, i would start at the pump, and water tank, and trace back every single inch of pipe, all the way to the back of the trailer, esp around the city water regulator, and the hot water heater.
If you have even the tiniest leak in the system somewhere, it will cause a pressure drop enough to kick on the water pump momentarily.
Grab you a flashlight, some patience, and a tall coolie
and start to tracing.... My money says you gotta miniscule leak somewhere behind a cabinet, or in the back of the trailer.
Keep us informed on the progress....(or the lack thereof....
Grosse Ile, MI