Jerry,
You can have one that looks like this but this newer version ups the pressure and the flow. Even when the campground supply is less than 65psi I have found that the flow at lower pressure is much improved. The older version is chromed plastic and looks the same but needs to be a 7222. They are cheap enough and I would change yours even if it was a 7222 to rule that out.
There can be several reasons the pump continues to run. One is it is bad and will not build pressure which does not seem to be your issue seeing it helps with flow. The cut out pressure switch could be bad. The check valve in the pump could be leaking back and losing pressure. The simplest issue that needs to be ruled out is to turn off the campground pressure feed, bleed off pressure in the coach using a faucet and then turn on the pump. Bleed air out of the faucets in the coach and shut the faucets off and then see if the pump pressure the lines up and shuts down reaching the cut off pressure. Air in the lines and pump can cause the pump to not build pressure and shut down.
If there are no leaks and this does not fix the issue then change the pump out. You do not want the old one to continue to run, it will burn itself out.
Later Ed