Big thanks for the clean-the-valve suggestion. I'm embarrassed that I didn't think of that one!
So, I cleaned the rest of the heads (which were very plugged) and the valve (which was fine) but nothing worked so I decided to start from scratch and check everything again. It turns out I now had two zones that caused the pump to cycle constantly and a third (of six) that would sometimes run smoothly and other times cycle. I know only one cycled in the beginning so I assumed I was dealing with a random problem.
I called two companies to come look at the only thing in common, i.e. the pump, but neither one of them called me back. I guess they have too much work this summer to give a poop about customer service and any future calls from me! :-(
So, I decided to do what I could. There was nothing obvious at the pump by the river so I looked at what I could with nothing more than a screwdriver. I did notice the contacts on the pressure switch were very badly burnt so I replaced it and so far everything seems to be working fine! Yeah!!!
That said, I don't understand why burnt contacts would make the pump cycle. I can see it preventing the pump from running but such was not the case. It was more like the pump would come on and somehow cause too much pressure, then shut off, only to have some water get through through the heads, the pressure drop, and then the cycle would repeat. But how could the pump randomly cause too much pressure or, more likely, a pressure switch randomly think the pressure was too high and prematurely tell the pump to shut off??? It's a mystery to me.
It's also a mystery as to whether or not the burnt contacts caused the problem or a mechanical problem in the switch caused the contacts to burn as they repeatedly opened and closed.
If anyone can tell me, I'd really like to know for future reference.
Thanks,
Audios