Still wrestling with this. To answer a few of the questions posed here.
1) If I open the whistling valve just a little bit, the whistling stops. There are two valves involved here. I close one and open the other to fill, then reverse the closings to run the pump.
The one I open to fill the tank is the one that whistles when it's closed. The one I open to run the pump is free-wheeling, but seems to work.
2) There is no water running from the city water fill. (I had hoped this was the issue, as it seemed an easy fix.)
3) The pump is a year old. As a test, I replaced it with an old pump I had (as a spare) and the results were exactly identical. Doesn't mean both pumps aren't bad, I suppose.
I'm wondering if pulling the city water fitting and plugging the inlet somehow would be a way to either confirm or deny a check valve.
Short of that, I think I'm looking at replacing the valves for filling/running water. I've been over every inch of the interior plumbing and find no water anywhere.
Thanks for the tips.