Fridge will run normal for 12 hrs or so, then shut off. Power is still on, though. If I fiddle with the thermostat/fridge control or tap the housing it will start to again. I beleive the thermostat needs replacement or maybe the defrost timer?

You still may be able to get a thermostat (i'm guessing that's what he's calling the cold control). Get your model number and hit the internet. Go page by page. It will be tedious but you may get lucky. Try every appliance part site that shows up. Sometimes o.e.m. parts sit in warehouses for years before they're sold if ever. You might find a brand new one. Also, check with your local appliance recycler. Sometimes they have good used parts.

A power outage is often accompanied by a voltage surge or spike. The modern fridge contains electronics and these do not like voltage surges so damaged electronics is a possibility though first it is necessary to discover whether power is actually reaching the inside workings of the fridge which means checking the power cord, connections, fuses, etc. with a multimeter.

My fridge is an older type without the fashionable electronics and when that refused to restart recently I gave the thermostat a sharp tap with a screwdriver handle. It started and mostly works normally but I bought a new thermostat as once it has caused trouble it almost certainly will eventually fail.

Check fuse in the plug use any other working one in the house to test it .
Hope that will help

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https://www.youtube.com/results?search_query=Why+Whirpool+fridge+not+maintaining+temperature+for+24+hrs%3F++Permalink

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If there is no cooling at all on the fridge section but the freezer works then it is confirmed that the compressor and gas pressure is OK. Now shut off the fridge and leave it powered off while you clean and check the evaporator for frost. If frost is seen it is a good indication as the compressor is working with sufficient gas within.
Now after about 6 hours plug in the fridge to start. Observe cooling to happen in about one hour and ice to from under three hours.
Earlier make sure that the thermostat was set to normal mode for freezer and fridge.
However if ice forms in the freezer then observe the fridge for about 24 hours for cooling to continue and the fridge to cut off . If not the auto defrost timer, or the thermostat must be checked. Frost formation without defrosting could block the air flow and so shut the air passage to the fridge. Also the blower fan must work and if this is not working then the fan can be faulty. Also a fault on the control unit is a suspect to non functioning of any one of these components.
If defrost timer works but frost keeps forming check while the compressor shuts if the heaters are working, check continuity of the heaters.
So try this test using a multi-meter on the defrost timer and the thermostat and make your observations so that we come to the solution.

Please check the link below for tips to remove the panels/parts:

http://www.espares.co.uk/advice/fridges-and-freezers/a/2

The fridge can cut off if there is a power flickering or variation as the relay will be energised to trip off. Once the fridge is off the self timer - 5 minutes- will be active to allow the gas/oil to come back to the compressor as the flickering would have produced an irregular pumping.
After about 5 minutes the processor will energise the relay to contact and start.
Now the main issue is the voltage in your household. if the voltage is low then the fridge will not start again as it will click and shut off. So make sure that there is sufficient voltage in the mains supply. Also the plug point where the fridge is connected must deliver good voltage/current as the wiring must be good to carry the load . If not the fridge will trip off whenever there is a power flicker.
Finall if the power is good then the fridge mut be checked as it cannot be off for 24 hours. register you complaint for warranty service.
If power is irregular use an auto- voltage stabiliser .

you moved the fridge out from the wall and it started running again?try opening the fridge door and tap on the fridge control and see if it starts up.i,ve gone into houses and then the woman tells me when the fridge shuts down she hits the side and it runs again.it's the control,this could be your problem also,when you moved the fridge the control could have kicked back on.let me know what you find

let it run 12 hrs for temp. balance.

Sounds like a thermostat got damaged when the power surged when utilities were restored. Thermostats are usually behind the temperature controller. Hope I've helped. BB

Bad defrost timer. Stuck in defrost mode.Turn the timer manually to reset. Possibly a bad cold control thermostat.(On/0ff switch)