After reading here, I decided to replace the water inlet valve. After over 15 years, I'm presuming the internals were corroded with deposits and that's what kept it from closing tight. Replacing the valve was relatively easy, particularly with the great video here. It's been over 24 hours and the problem seems to be fixed.
Two tricks I used: 1) piece of cardboard under the valve after replacing to easily see if there were any leaks - they show up quickly on the cardboard, and 2) had two cell phones, used one pointed at the valve to FaceTime the other one so I could see if there were any "catastrophic" leaks while downstairs turning the house water back on.