Either the defrost circuit has failed or the refrigerant is low - my money is on the defrost circuit.
Most defrost circuits have three main parts:
1) defrost electric heating coil
2) defrost terminator
3) defrost timer.
The heating coil and terminator are in the freezer compartment - behind a protective panel. The heater is usually piggy-backed on the freezer coil and the terminator is in contact with the freezer coil to detect its temperature. The defrost timer is a simple assembly of a clock motor with switch contacts that can be located anywhere the manufacturer desires. The timer turns on or enables the defrost circuit every 8 or so hours for up to 30 minutes give or take.
When the timer has enabled the defrost cycle, the cooling mode ceases; the compressor shuts off and power is sent through the terminator to the heater. The heater warms and melts any ice build up on the freezer coil. The water drips to a pan and flows down the tube to a pan under the fridge - where it is evaporated off. The heater warms the freezer coil until either a) the timer returns to cooling mode or b) the terminator senses a preset rising temperature on the freezer coil. Once either condition is present, power is interupted and heating stops. The compressor is energized through the adjustable thermostat in your fridge. Since it is warm, the compressor turns on and cooling begins.
If any of the components listed above (1,2 or 3) have failed, the defrost cycle never warms the freezer coil and the ice never melts to clear the freezer coil as intended. Air can not be circulated through the freezer coil since it is choked with ice, so even though the compressor runs, the fridge and freezer spaces never get colder. The adjustable thermostat never sees the temperature you've set so the compressor never shuts off.
Water dripping in the fridge is melting ice from the freezer space - as the freezer is not getting colder - only the protected space around the freezer coil is.
To fix this, you'll have to get the service manual or schematics for the fridge to determine where the parts are located and do some troubleshooting with a multimeter after disassembling the freezer compartment and wherever else to access the timer if needed. This is not a good first appliance repair job for a DIYer due to the danger of refrigerant and testing live electrical parts in closed in spaces.
I hope this helps.