I noticed Thursday that the ice maker in our model year 2000 Amana side by side refrigerator wasn't keeping up with our daily use. The ice bucket is typically full at all times, and it was only about half full and most of the ice cubes were stuck together in a solid mass. Today(Saturday), there was a puddle of water at the bottom when I opened the freezer, and the ice cream was soupy. There were still ice cubes, however, and meat still appeared to be frozen solid.

When I first discovered the issue this morning, I emptied the freezer and removed the inside panel. There was no ice at all on the coils and it didn't seem that the fridge was working at all. After a short time, however, I noticed that the fan began to run. Now there is a thick layer of ice on all of the coils, the fan and condenser are both running, and the condenser is hot to touch. Any idea what could be the problem?

Any help would be greatly appreciated, thank you!

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When I first discovered the issue this morning, I emptied the freezer and removed the inside panel. There was no ice at all on the coils and it didn't seem that the fridge was working at all.[/COLOR]
You were probably in a defrost cycle. This shuts off the compressor and both fans.

After a short time, however, I noticed that the fan began to run. Now there is a thick layer of ice on all of the coils, the fan and condenser are both running, and the condenser is hot to touch. Any idea what could be the problem?
I am not sure this is a problem.
Once a defrost cycle ends the compressor and fans come on.
They will run until the unit reaches the set point temperature and then will cycle to maintain that temperature. The freezer should maintain at 0 to 5 degrees F. Since you have a ice maker it should be closer to the zero.
Perhaps put a thermometer in the freezer so you can tell what temperature it is at.

Since you opened the unit up it will take quite a while to get back down to set point.
Usually the evaporator/freezer coils just get a layer of frost not ice but this may be related to your location re: the ambient humidity.

I am not sure what your original problem was but it may be related to defrost.
The way defrost works is that every 8 hours of compressor on time the unit goes into a defrost cycle. The cycle is about half an hour and is a fixed time.
The compressor and fans shut off. The defrost heater comes on and melts the frost/ice from the evaporator/freezer coils. This water should then go down the drain tube to the drip tray under the unit.
Now lets say all the frost is melted off at 15 minutes then the defrost thermostat opens. It opens just above freezing. This kills power to the heater so that the defrost cycle will continue with the heater off. This prevents excessive heat build up in the freezer.
It could be that the defrost thermostat did not open.
When frozen it should be 0 ohms.
At room temperature it should be infinite ohms.
If it is deformed/bulged then replace it even if it does measure OK.