[DickR]

If it keeps running, with too low a temp in the fresh food part but not cold enough in the freezer part to shut down the unit when it gets cold enough there, then my first guess would be an iced-up evaporator coil. I have had this happen to my Kenmore side by side a couple of times over the last few years.

That coil is in the freezer compartment. It will be protected by a metal or more likely plastic panel. There will be a fan under there as well. You'll have to remove the frozen food and any shelves in the way, plus maybe the lightbulb and any plastic cold air routing panel above the coil if the fridge is a side by side. The removal of stuff may take you a few minutes to figure out, but it may be just a few screws and perhaps popping off a press-on panel. Turn off the unit, of course.

A giveaway clue is if you see frost or ice covering the panel over the coil. That usually means direct contact between the panel and the coil by means of ice buildup. You may have slots in the covering panel that let you see part of the coils themselves, so a buildup of ice may be apparent.

What would lead to this is one of several things. One, the defrost timer could have failed, so the thing never defrosts itself. You could have something inside preventing the door from closing all the way, so that the freezer is acting like a dehumidifier for the whole room. If that's the case, the condensate freezes instead of dripping off as liquid water. Or, you could have put some hot uncovered liquids into the other side, and the vapor came out as frost on the evaporator coils. I've wondered if an auto icemaker can do this also, as it exposes additional water to the air circulating through the unit. Overloading the coils with frost can result in the inability of the defrost cycle to melt it all off in the time allowed by the design. Then you just add to the buildup with more frost as the refrigeration cycle restarts.

If there is a buildup of ice clogging the evaporator coils, the first thing to do is remove it. The slow way is by leaving the freezer door open and letting it melt off as the unit warms up from room air. You can speed things up by setting a fan on the floor or a chair. The turbo approach is to use a hair dryer; that can do the whole job inside of an hour. You'll have to sponge up the melt water as it drips down. I'd suggest covering the outlet from the drip tray under the coil with a sponge or cloth and let it hang over into a container lower than the tray.

Edit: don't try to pick off the ice pieces with a tool or your fingers. The aluminum fins on the coils are very sharp! Let the melting do the work. Some pieces of ice may drop off on their own; those you can pick up and toss. Just keep the fingers off the sharp fins.

After the ice removal, reinstall the panels, shelves, lightbulb, etc., and turn the thing back on. Give it a few days of running to see if the defrost cycle is keeping the coil free of buildup. All this assumes you found ice buildup on the evaporator coil, in which case the problem is the defrost timer or heating element it runs, or else just a temporary overload that your manual defrost effort has got you around. If the auto defrost is the problem, you'll know in a few days or so of running and the original symptoms return.

If ice buildup was not the problem, then then you could have a leak of refrigerant, so that you get some refrigeration, but not enough to make the freezer part cold enough to turn off. If this is the case, eventually you won't get enough refrigeration even to keep the fresh food side cool. This you can't fix on your own.