The icing is caused by the fridge evaporator. You have a defrost problem.
The following describes the freezer defrost system but also applies to the fridge side.
The evaporator coil behind the cover on the back wall inside the freezer will ice up under normal conditions. Every 8 to 10 hours for around 20 minutes the defrost timer (or in most newer models the electronic adaptive defrost control) will turn the defrost heater on to melt the built up ice. There is a defrost thermostat which prevents the heater from overheating the freezer by breaking the heater circuit when the temp reaches close to 32 degrees F. The entire cooling system shuts off during the defrost cycle and starts back when the timer advances through the cycle.
If this ice is not melted it will continue to build up until the air canít flow over the coil to circulate the cold air through the freezer and into the fridge. The temperature change in the fridge is usually noticed first followed by the freezer.
If the defrost thermostat is bad, it can prevent the heater from coming on OR it wonít turn the heater off when it gets too warm. It is clamped to the evaporator coil at the top to sense the temp. If it appears to be misshapen it is bad.
With an ohm meter it should show continuity when cold and none when warm.
You can also bypass(disconnect the two wires plugged into it and twist them together) the thermostat to see if the heater comes on then. If it does then you know the thermostat is bad and needs replaced.
The defrost heater is located on the evaporator. It is in a tube which is at the bottom and can also go up the sides of the evaporator. On some types you can see a burnt spot if itís bad. With an ohm meter it should show continuity from end to end when disconnected from the wiring in the freezer. You can also test the wiring for voltage when itís in the defrost mode.
If you have a defrost timer you can test it. It can be located under the fridge behind the kick panel on the front. Some are in the fridge with the controls at the top. You can turn the defrost timer till it clicks and everything shuts down. The heater should now come on. If it does, replace the timer because that means the timer is not running. If it doesn't, check the heater and defrost thermostat. Turn the timer again till everything starts back up to end the defrost cycle.
If you have an adaptive defrost control instead of a timer, replace it if the heater and thermostat test good. It is located in the fridge with the controls in some models and on the back in others.