The evaporator fan should be tied into the cooling cycle, so it makes sense that if the thermostat is calling for cooling, everything involved in that process is operating. 

I assume that when you turn the cold control to "OFF" everything shuts down.  If not, then yes, there's an issue with the thermostat and it's probably not something that can be fixed.

There are universal aftermarket cold controls out there, and the best place to find them would be on ebay.  You'd have to extract yours or take a good picture of it so you can compare it to what you might find on line.

I had a similar problem develop on my '57 GE Combination.   The fridge would run continuously even on the warmest setting.  I pulled the cold control and there was nothing visibly wrong, but  I tried to take the assembly apart and it did not turn out well.  There's nothing serviceable inside.

The one thing I do recall is that the NOS replacement control I installed had instructions and adaptive mounting hardware for other makes besides GE, and Frigidaire was one of them.

Let's hope it's not a tired compressor.  I hadn't thought of that possibility, but yes, if it's getting overly cold in the cabinet, then the compressor is likely still OK.

The tubing is probably either clipped into place, or like my '57 GE, has a couple of clasps that are held by small screws (see picture).  You may need to remove the rear panel to get at the evaporator, but I'm going on fuzzy memory of what these Cold Pantry models look like.  If you can post a picture of the interior, that should help.

Bummer.  It's hidden.  If there's no way to view from the top (like below the freezer section, I'm assuming), I don't know how you'd fish through a new one.  Worst case scenario would be to securely attach a line or thin wire and pull the thing out from the front end, and use that same line to pull the new one back through, but there's no telling whether you'd hit a snag trying to remove or reinstall.

Others here are more familiar with Frigidaires than I am, and may be able to provide information on the R&R process.

Tie a string to the end of the sensing bulb on the old thermostat.. When you pull the thermostat out, the string will come with it. Then untie it, and tie it to the end of the new sensor. Pull the other end of the string, like a fish tape. Kinda threads it behind the liner.

Did the temp thermostat attach a sensing lead to the cold plate? Or was it one that sensed air temperature?

So did it need the new thermostat after all?   How did you determine that?  Just curious.

Can you use something like a length of coat hanger wire instead of a string?  If that will push through to the rear opening,  you should be able to pull the capillary tubing behind it.  How you keep it attached to the wire is the tricky thing.

Removing the liner could be a big job, but it can certainly be done.  I wonder what a repair tech would have done back when these refrigerators were daily drivers in homes across the nation.  I can't imagine it would have been a lengthy process to replace this part.

Thank the stars the coat hanger worked! I can't imagine pulling the liner, or if it could even be done.

Give it some time.  I don't know if there are system design quirks on these refrigerators that may cause them to operate differently from other makes.    For sure if yours is working at bringing the cabinet temperature down to the 37-40 range, it could run for a while.

Do you have the cold control on the warmest setting?  That would be a good way to determine whether there's another problem, although I can't imagine what else would cause the compressor to keep running if both the thermostat and relay have been replaced.   Do you hear a click when you adjust the cold control lower or higher?   That would indicate that the contacts are opening and closing properly. 

Perhaps it's a wiring issue.  Do you know whether anyone has messed with the relay or thermostat before?

Just bumping this one with hopes for an update.

I'm stumped too.  Hard to say if it's the thermostat or the relay.  What we need is a wiring diagram that's color coded.

Wires on the control shouldn't matter, just 2 of them I believe. It still sounds like a control problem, the one you're using is known good?

Interesting that it can still get below freezing inside and yet the evaporator accumulates no frost, but it does make sense that with the evaporator not getting cold enough, the thermostat registers that the machine needs to keep running.

Let us know how it goes after the recharge, and let's hope the refrigerant leak is a very slow one.  Freon is not cheap, nor is it good for the planet to release it into the atmosphere at any rate.

Fingers crossed.

Ken, I agree.  Stan's monitor/box top seems to need a recharge every dozen or so years.  Let's hope the Cold Pantry's leak is as slow or slower.