There are no tests you can really do with the board out unless you have a load jig to simulate loads. That is your adaptive defrost control (ADC), it could be sticking in defrost or in the steps of coming out of defrost.

Reassemble the control panel and leave it dangling so you can get to it. If it runs after plugging it back in let it go, when it has stopped running for more than 60 minutes (to allow for the chance of it being in defrost) tap on the ADC with a nonconductive item and see it it restarts. Repeat for thermostat/cold control if not.

That is not a definitive test but a decent one. For more clarity you'll need to do a little testing while energized since you can't reliably test for resistance/continuity while energized you'll be using the AC voltage setting on your meter. When it is warm but should be running test across the thermostat for potential V AC, meaning if it is closed there is 0 potential and if it is open you'd see 120V between the 2 terminals on it. Closed means it is asking for the unit to run, open means temps are low enough for it to be satisfied and no power is sent to the compressor/fans, etc....

Another test for the thermostat is to jump across it with an insulated tool or wire while it isn't running, assuming it isn't in defrost at that moment; the unit should start if the ADC allows it.

Either could be your problem but odds are better that it's the ADC.

The ADC measures compressor run time and possibly door openings to determine when to go into defrost, they can fail to restart the machine after defrost. You description of it running for a while then failing to stay running points towards the ADC also. Unplugging it for a while resets the timer in the ADC so it starts fresh and begins to accumulate time until it wants to defrost again.

If it has developed a full frost pattern on the evap the sealed system is probably fine. That's the big money repair that is not DIY. The defrost thermostat is cheap. The ADC is more but less than a new unit.

Personally I'd keep an older functional unit unless the repair cost is more than 1\2 the cost of a similar new one. Even at that point the quality of new units is low enough that I might still repair an older one, that's what many others in the field say also.
Ultimately it is a gamble, at 13 years old other issues become more likely, however a new one will probably not last that long. My home fridge is 20 years old and I'd trust it more than many new ones.

As for selling it I believe in backup refrigeration if you have space in a garage or other area, it's nice to have extra space and and if the new one fails young you're not without one entirely.

So the fridge is still staying cold but I've noticed it runs almost constantly and the fresh food side is right around 32 degrees with the thermostat in it's normal "starting position." With the unit running, I tapped the cold control knob with a screwdriver handle and it cut off immediately. I'm now thinking the issue is as simple as a faulty cold control. Seem reasonable?

I installed the new cold control/thermostat 4 nights ago and everything has been running perfectly since. I have a thermometer in both the freezer and fresh foods side and the temps are staying nice and steady.

That's it - just wanted to post up that it's fixed. Thanks for all the help and advice!