If the freezer is maintaining 0 degrees but the ice cubes are melted/not frozen completely then I suspect the problem is the the ice maker.
The ice maker has a built in heater that is suppose to only stay on long enough to melt the cubes from the mold so they can be harvested.
From the description you give I suspect the ice maker element is staying on to long. There is a thermostat in the ice maker that might cause this fault.
You can try to confirm the freezer is staying at 0 degrees. There is a defrost system that uses a thermostat to shut off the defrost heater before it gets to hot. If the defrost thermostat failed then the defrost heater might stay on and cause the ice to partially melt. You can check this by placing a couple of ice cubes in a small bowl in the freezer. If the cubes melts during the cycle then there is a problem. You can place one bowl near the ice maker and you can place a 2nd bowl in the lower part of the unit.
If the freezer isn't maintaining temperature, this could be the issue. This could be caused by several different problems.
Dirty condenser coils below the refrigerator could be contributing to this problem. I recommend checking and cleaning these coils if necessary.
Check the door seal on your freezer. A leak through the freezer door seal may be allowing warm air to enter the freezer and melt the ice.
Make sure that your temperature control is set properly.
If these tips do not help resolve the problem, it is likely that you have a component failure or a sealed system problem. The thermostat could be bad. You could have a condenser or evaporator fan problem. You could also have a defrost system failure that is causing your cooling problem. A weak compressor would also cause this symptom. I recommend calling a service technician to diagnose and repair your cooling failure. Service can be scheduled at the following link:
Sears Home Services
I really hope this information helps you today. For further assistance, please respond to this thread with the complete model number and more detail.