So I resolved to fix this problem. I first tried taking the spray unit apart to make sure it wasn't clogged. It wasn't. Even with the spray head removed, half the water still came out of the faucet.
Next--CLEVER ME--I searched the Internet for a description of the problem. And I found several dead-on hits indicating that the culprit was a worn piece of hardware called a "spray diverter." I printed this information and drove to Lowes. After much searching, I found a spray diverter that was described as a replacement for Peerless faucets. I checked on complete faucet/spray units, and the cheapest acceptable one sold for $68. "Ha-ha," I thought. "I'm going to lick this problem with a piece of hardware that cost only $2.79."
But there were some obstacles. Taking apart the existing faucet proved to be non-intuitive and difficult. When I had disassembled it as much as I could, I still couldn't get to the diverter. With a completely removed faucet, which looked grotty as hell, I figured I might as well go back to Lowes and get a whole new faucet sprayer.
I returned to the store and opened the box of the $68 Moen unit. The pipes extending from the faucet did not seem long enough to me. I looked for a Moen with longer pipes, and chose one for $88.
Installing the new faucet was not all that difficult, although getting the pipes lined up enough for the threaded connectors to match took some time. When I had it all together I turned the water pressure to the house back on and saw little streams of water leaking from the connections.
Additional tightening with two wrenches eventually did the trick, and now, after a couple of hours, I have a nicely functioning kitchen faucet and spray unit. And a spare spray diverter, if I should ever need it.