Fridge coil won't get cold
Defrost cycle. Your defrost timer, element or thermistor are probably bad. If you unplug it and after it has warmed and defrosted, then you plug it in and it works for what seems to be a day, then ice builds up again, then its not defrosting as it should. This ice build up will block the fridge vent and not allow the fridge to cool. This non-defrost will make the freezer think it has cooled to the temp you selected because the thermistor is covered in ice. Its hard to figure out which part unless you have the tools. A multimeter would allow you to test the defrost element and the thermistor. Just your ear and hands would allow you to check the timer (assuming it is a manual timer) by manually rotating the timer to different spots. Make the unit click on, run for a bit, listen and feel for cold air. Turn it and make the unit turn off. If you can do this successfully, the timer is most likely ok. Another test to check the timer, is to move it just before or after the flip on/off time, and let the fridge do it by itself. If it does, then the timer is ok. 

Once you have verified the timer is ok, you can test the other two with the timer. Set the timer to the defrost cycle and let it run a complete defrost cycle. Check for water and ice in the areas you found them before. If present, then the element is not defrosting/melting the ice. Move on to the next test.

The thermistor is a tricky test, I had a bad one, but it never tested bad. You can take it out, place it in warm water and get an ohm reading, then place it in ice cold water and get a different reading. Compare those to the specs listed for that thermistor. Out of spec means the thermistor is bad. I would bet that the thermistor is most likely your culprit. It is what tells the defrost element to turn on or off to defrost and melt the build up of ice. 

Before you test the element ,if you have a bad thermistor, replace the thermistor and run the unit for a day and monitor it. Do the timer test to see if it is now defrosting. If not, then continue to the element test. 

The test on the defrost element, you do need to remove it or isolate it from the rest of the fridge usually by unplugging it, you use the ohm setting on your meter again, and test for continuity. Open circuit, bad element, closed circuit, good element. 

I would do these tests in this order as the parts get harder or more complex to remove or find.