My Greenway water cooler/heater does not cool the water anymore. The on demand heating system works fine but the compressor runs constantly and the water remains room temperature.

If your Koolatron cooler is not cooling down, even though the fan is working, there may be a problem with the compressor or the refrigerant.
The compressor is a critical component of the cooling system, as it is responsible for compressing and circulating the refrigerant that absorbs heat from the air inside the cooler. If the compressor is faulty or damaged, it may not be able to compress the refrigerant properly, resulting in poor cooling performance.
The refrigerant is a chemical substance that absorbs heat from the air inside the cooler and carries it to the compressor, where it is compressed and released. If the refrigerant level is low, the cooler may not be able to absorb enough heat to cool the air effectively.
To fix this problem, you will need to diagnose and repair the faulty component. This may require the assistance of a qualified repair technician, as repairing a cooler's cooling system can be complex and hazardous.

PERHAPS A NEW SYSTEM IS NEEDED .

Many newer units are considered “fully evaporative.” This means the water accumulated from dehumidifying is actually discharged out of the exhaust. But, many times the evaporation can not keep up with the cooling demand. The machine has to stop running for a bit to get rid of excess water so it doesn’t overflow. Try hooking up a permanent drain!!

Small refrigeration units like this one like to fail to cool by developing a small leak and slowly
leaking refrigerant.
THERMOSTAT FAILURE CHECK:
A technician will check the thermostat using using a DMM (digital multimeter). But you can suspect the thermostat if you set the thermostat high and the compressor motor does not run,
then set thermostat low, and the compressor motor also doesn't run - in other words, the thermostat is failing to start the refrigerator's refrigeration system no matter what its setting - an obvious failure of the thermostat.
COMPRESSOR FAILURE CHECK:
If the compressor is dying, you would hear the compressor running almost constantly, but not
sounding quite right (uneven noise, clunking, tapping).
REFRIGERANT LEAK CHECK:
If you unit is low on refrigerant (due to a leak), the compressor motor will be running
almost constantly, but it will sound Ok.

When empty of water remove the back panel and the cooling fins are most likely covered with an inch of dirt! Find an air compressor and blow out the dirt. It worked on mine!

Change cold control

You definately want the compressor to shut on and off with the demand set by the thermostat. It is no different than a full size home split heating and cooling system. You set the thermostat for the temperature you want and then you let the system decide when to run the compressor

alot of time those small units do not have a fast recovery rate so if they are on for a min or more they might start to lose temperature. if it's on demand especially in cooler climates it might not have enough power to heat it up and keep up with your demand

I have a hamilton beach water cooler with storage cabinet # 87330. The cold water is not cold and the cooling light stays on. What is the problem and how can I fix it? Thanks!

What you're likely referring to is the blowing air starts and stops erratically. Since the heater uses heat circulated from the engine, if the air is constant and the heat comes in and out then that's a different issue.
Firstly, if the airflow starts and stops, you'll find a loose connection between the the fan switch and the blower itself. If the connections are sound, then look for a loose ground connection to the blower. If all electrical connections are good, and it still starts and stops erratcially, then it's time for a replacement blower.
Secondly, If the air is constant and the heat comes in and out, the the options are a faulty water pump or partial blockage in the colling system or heater core. If the water pump is failing and working erratically, then when the heat dissipates in the cabin airflow (it has used all the heat in the heater core and no circulation to replace the cooled water with hot water) the dash temp gauge should start to climb, indicating that the coolant is no longer circulating, the heat has left the heater core but the engine is not being cooled. You'll need to replace the water pump. If this is the case, take care of this ASAP as you run the risk of a an engine overheat.
If the dash temp gauge looks normal, then you're likely looking at a partial obstruction in either the cooling system or the heater core that blocks the flow and then works its way loose erratically. Limescale and crud buildup in your cooling system (especially in a 15-ys-old car) can cause this when flakes of buildup brake loose from the walls of the larger channels and get caught up in the smaller ones. Your local parts store will have additives to clean out the cooling system, but if they don't work (they can only do so much) then you're looking at a professional cooling system flush. If the issue persists after that then you're likely looking at a heater core replacement.
Also check to see that your a/c is not kicking in erratically. It shouldn't completely cool down the full-hot airflow, but it will cool it. You can tell if the a/c unit is kicking in as you'll hear the compressor clutch kick in and the engine revs/sound will change under the load. If this is the case look for a short in the a/c wiring.
Hope this helps, and please rate my answer.