Is the one you're having a problem with a frost free unit that's frosting up? You said ice on the coils. That shouldn't happen. A frost free fridge will frost up for a few reasons. Do the doors seal up well? If not you're allowing too much moisture inside the unit. It'll freeze up. There should be a fan running inside when the compressor is running and the door is closed. Check it by pushing the door switch closed while it's running. Same switch usually turns the light on and off. Now the fan might have ice blocking it too so the next thing is checking the defrost system. If the fan is bad gotta change it. Gotta have airflow over the coils. Easy enough and not expensive. As to the airflow thing, if there's food literally packed in the there it may be blocking the airways. The fan has to move air around the inside of the the thing it it's gonna freeze up as if the fan were bad. Make sure the fan is moving air.
Frost free fridges have a timer that turns the compressor off and sends power to a little heater that is on the cooling coil in the freezer section. The number of times it'll run in a day varies but they all defrost at least once a day. Find the timer. Sometimes they're in the open in the bottom section. Other times they're somewhere near the thermostat in the refrig section. There's usually a little plug covering a hole that gives access to it. Look for something like that and pry it off. It should pop out easily. Or just look for a hole that looks like a screwdriver would fit into and look for an odd looking plastic shaft inside. Odd because it's shaped so a screwdriver can only turn it one way. Once you've found that, slowly turn it clockwise with a screwdriver. You should hear a series of clicks ending with a louder click. Stop turning. The compressor and fan should be off. Close the door. Give it a few minutes and look at the drain pan in the bottom section and see if water is dripping into it. It should. That will indicate the timer is sending power to the heater and it's melting the ice. If it isn't, then you're going to go a step further. But first let's see if the timer is advancing. After about a half an hour or so after you started the defrost cycle the timer should advance, you'll hear a click and the compressor will begin to run again. If it doesn't, the timer is bad and likely your problem as if it isn't coming out of defrost that means it isn't going into defrost either! It's stuck. Little plastic gears inside and one has lost a tooth. Gotta replace it. It just unplugs and is held on with a couple of screws. Easier than you'd think.
OK next step. (you could actually do this first. Wasn't sure if you had access to a meter like this.)You're gonna need an electric meter with an inductive clamp on it to read the amperage the fridge is using at a given time. It's the easiest, most non-invasive way of doing this! Back underneath, look at where the power cord goes into the unit. The wires will separate there. Put the clamp on the hot (black) wire and set the meter to read amps. You're not handling any bare wires here, its perfectly safe. While the compressor is running it'll show the amperage the unit is drawing. It varies by unit but you're looking for a number that ain't zero! OK now go to the timer and advance it into a defrost cycle. With everything obviously stopped you should still be showing amperage. 3 or so depending on the unit. Anything but zero! Keep a watch on it and it should drop out after a few minutes. Why? Because there's a little stat on the coil inside that stops the heater at about 38 degrees. Enough to defrost the coil but not your food. And then after a while more the timer will put it back in cooling mode.
Now if while in defrost and you're showing amps drawn by the heater and there is no water dripping into the pan that means that drain tube is blocked. Gotta clear that so the water is removed otherwise it just refreezes. It starts in the freezer section under the coil. It could be blocked with ice. Slowly pour some really hot water in it. Should end up in the drain pan in the bottom.
OK you've done that and the heater shows no amps. Problems include a bad heater coil--duh! Or that little stat I was talking about is bad and not allowing power to the heater. Or the contacts in the timer have gone bad and while the turny roundy part is working, its not sending power to the whole defrost system.
The only way to troubleshoot this further requires taking apart the cover on the freezer section to get access to the heater and stat so they can be tested with a meter. If it's a side by side unit--good luck with that! Prepare to have carpal tunnel by the time you get it apart! I don't know how comfortable you are working with electricity and meters but I can walk you through it if you want.
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