After silencing the alarm, it seemed to be running as normal without any unusual noises, but food was starting to thaw from the top shelf on down and condensation was dripping from the roof. We were able to place most of the food in 2 large ice chests and then the rest in the freezer compartments of 2 fridges we have in the house. I then turned down the temp dial to the coldest setting and let it run for an hour or so; no change.
After researching on the internet, it sounded like our symptoms could be pointing to a defrost timer issue. I pulled the freezer away from the wall, located the timer and used a large screwdriver to slowly advance it. While doing this, at one point I could hear the compressor kick in. I pushed the quick freeze button and let it run for a while. When I checked it in a few hours, the alarm light had gone out and water that I had placed inside was frozen. We took a chance and put the food we placed in the ice chests back in the freezer. Everything was fine several hours later.
I ordered a new defrost timer, it arrived in a couple of days, and replaced the old one with it in a matter of minutes by disconnecting the wire harness from the old timer, removing the 2 screws holding it onto the bracket, screwing the new timer to the bracket, reconnecting the wires harness, and done. It's been a week or so and no problems.