With the newly gained knowledge from this forum, I spent some time this evening trying to figure out the mysteries of my hot water heater. I checked for loose/corroded connections and everything seemed good. I turned on the hot water heater and the burner ignited right away as always. The piezo ignitor remained clicking as it has in the past. Upon further looking, I found that the end of the piezo ignitor was too far out of the main flow of the flame. I gave a gentle push on the bracket holding the ignitor, and the sparking stopped. The picture shows the position of the ignitor after I made the very minor adjustment. I suspect the electronic module is still sending a signal for a spark, but it cannot occur when it is in the flow of the flame. To check my theory, I shut off the gas valve at the propane tank while the burner was on. As the pressure died down, the flame went out and the ignitor immediately started sparking. As soon as I turned the propane valve back on, the flame re-ignited. One problem solved!
I then left the hot water heater on so I could get a temperature measurement as Joe suggested. I used an infrared thermometer I purchased years ago that I have found to provide reliable temperature measurements. It took a while but when the burner turned off due to the thermostat, I checked the temperature. I did this by filling a cup of water from the kitchen faucet and taking a measurement. It measured 152 degrees. Too hot for us! I have the adjustable thermostat on order, it should arrive next week.
I hope this information is helpful to others. Thanks again for everyone's great advice!
JJ