Ok - here's what I've done so far. Shutoff campground water and relieved pressure. Cycled pump a dozen times to lower water in tank until it quit overflowing. Then I left the pump switch on, and I didn't use any water. It comes on about every 20 minutes, so it's leaking internally.
It's a Shurflo 4008. Here's what the manual says "Lack of sanitizing is the number one reason for premature pump failure and poor performance over time. Lack of sanitizing will cause scale build-up on the diaphragm and valves, causing low flow and leak back occasional pump cycling with no faucets open or tank filling up when hooked up to city water".
Guilty as charged, but in my defense it's only 10 months old and it's been used regularly. When I get back home, I'll have to see if the dealer will take care of it under warranty, or I'll have to take it apart myself.
For now I guess it's not really hurting anything.