That did it! It was the diaphragm. Your advice of attaching the new diaphragm in the bonnet first made the assembly a piece of cake. I just put one screw through a little on each side, lined it up over the bottom, continued to tighten those two screws, then put the additional 6 screws in. No leaks and it worked immediately.
Upon questioning the clerk at SprinklerWorld again about it being the controller, he said some Rainbird controllers are known to have a problem that causes a pulsating chatter. I have a Champion unit. And I think what I called "pulsating" was just a very weak flow causing the heads to momentarily come up then drop repeatedly.
In any case, again, thanks for your words of wisdom.