I thought that I'd managed to solve the problem of the toilet running on when I made the "adjustment", but when I was in the bathroom today, I thought I heard water running where it shouldn't be and -- sure enough! -- the valve was still not shutting off before the water gently overflowed the top of the fill pipe. Not only is this a waste of water, but it's a waste of money.
When I'd had the gusher, I went out a few days later and bought a new valve, intending to install it when I had time and a good reason. Today after work seemed like a good time.
And about forty minutes later, the new valve was installed, the mess cleaned up, and the toilet no longer runs on.
This is progress.
Time to order the pizza...