The valve was leaking and I decided I'd give it a shot.
By the time I gave up, the screw was stripped, but I managed to fix the leak by manually opening and closing the valve.
Within a week, I turned off my sprinklers for the season.

I'm afraid that it will leak again when I turn the water back on. I absolutely will not cut out and replace a valve. The valves are very close to my house and I don't want to risk creating a flood around my house's foundation. I'll end up paying hundreds of dollars for someone else to mess with it and accept it as a bargain.