It sound like the fridge may not be defrosting or the fan in the freezer is not working.
If the fridge is not defrosting the air flow will be blocked and not able to reach the fridge section.
From what you have said about it working after you had it unplugged it sound like a defrosting problem.
Check and see if there is an ice build up behind the back panel in the freezer section.
If there is locate the defrost timer in the control panel in the fridge section.
With the compressor running turn the little knob on the timer until the compressor shuts off.
Watch and see if the ice melts, this should take about 20 to 25 minutes.
If the ice melts you will need a new timer.
If it does not melt but the compressor starts back up you have a problem with the heater or the defrost thermostat.
These are locate in the freezer section on the coils behind the panel where the ice is.
You will need a meter to check these.
Here is a link to your fridge model 596.66992400.
The part number for the defrost timer is PS649809.
Your fridge is made by Amana for Sears. | 
