A failed start relay would not start a compressor unless there was absolutely no load on the motor but then very very rare. If the compressor is running be isn't cooling it means your compressor there is no load on the pump and it's broken or refrigerant has leaked out.

But do you even know how your twin cooling unit works? I can't see it having a single speed induction motor with two separate loads without being a dual speed motor in inverter motor. You need to know what you have and the sequence of operation to troubleshoot it other wise you're just guessing.

A model number might help someone willing to give you better suggestions so they know what you have.

Thanks Kevin. The twin cooling system on the Samsung RF267ABBP refrigerator uses one compressor and condenser along with a separate evaporator in the fridge and freezer. The fridge is first in line. This happened on June 10 and hasn't changed since then. The fridge stays around 47 degrees, and the freezer is around 78. Again, no continuity on the 10-ohm rated start relay, which tested at 1.25k ohms. Yet the compressor starts and stops, keeping the fridge relatively cool. The compressor use an RSCR motor, but is acting like a PSC motor running with only a run capacitor.

Start relay: The start relay used is electronic which uses a thermistor so it should not have electrical continuity. Ohm reading s/b around 1.3K so yours is fine and it's working as designed, it is not the problem.

Running but shutting off: The compressor runs a long time but turns off without cooling the freezer. The only thing that will turn on the compressor is a call for cooling. The only thing that will turn it off is meeting temperature set point, a call for defrost by a temp sensor, a default timer programmed in the PCB ,or the compressor overload protection. There is no other monitoring devices in the refrigeration system to the PCB to turn off the compressor. It's a simple common refrigeration system and once called for cooling it runs until temperatures are met. There Is nothing wrong with any of the electrical or electronic controls or controllers. The problem is only in the refrigeration system,

Your compressor runs but but doesn't cool: Yes the refrigerator cools a little but the compressor isn't moving enough refrigerant through both evaporators to cool correctly. The compressor and refrigeration system is on it's own to complete the refrigeration cycle and this is where your problem is. So forget about any of the electronics, controls or sensors because they are not the problem and why diagnostics show no trouble codes.

Diagnosis: The problem is not enough refrigerant moving through the system. knowing the details I would rule out a leak. A slow leak would not blow the refrigerant charge in one day but still allow the refrigerator evaporator to continue to slightly cool nearly a month later,

See why the compressor shuts off: Determining if it's a defrost cycle or the thermal overload using a simple voltage test. If the overload has a 120 volt drop across it the overload is a open circuit, and the compressor will be hot. A defrost cycle would show no voltage between wire S/BLU and ground.

Tripped overload: This will suggest a heavy load on the compressor or a worn out compressor motor. Check current draw from the compressor and compare to the running amp rating stated on the nameplate of the compressor. (not the refrigerator). If it's high the motor is shot or a heavy load is on the compressor. The reason for a heavy load needs to be determined, but since you eliminated all the reasons I believe it's the defrost and not an excessive load.

Checking compressor valves and liquid line restrictions: While testing current draw note the current. If the current is Low it will indicate the valves in the compressor are shot, or a restriction in the liquid line. A restriction will cause the evaporator, compressor, and condenser to be starved of refrigerant and it will accumulate in the bottom of the condenser. The restriction could be anyplace from the condenser to the metering device, (capillary tube). The drier is the most common restriction point followed by the metering device. The only way to tell if there is a restriction is a cold or frosted spot after the restriction. A Liquid line restriction is often misdiagnosed as low refrigerant. My top pick is bad compressor valves, turning off condenser fan and wrapping up the condenser (just as a dirty condenser with a dead fan would do) would tell you if the valves were bad if low current didn't come up and the compressor doesn't get hot or even a toasty warm.

Kevin, thanks again for all your input. I really appreciate it.

I'm not understanding your initial comment regarding the 10-ohm PTC start relay. If it isn't supposed to have continuity across its two connectors, how would current pass through the relay to briefly energize the start winding before the thermistor heats up and effectively opens the start circuit, de-energizing the start winding? Also, doesn't a 1.25k ohm reading on a 10-ohm rated PTC relay indicate it is faulty? All the videos and instructions I've read on testing these PTC thermistor relays state that when the relay is at room temp there should be continuity across the two terminals, and that a very high ohm reading indicates a faulty relay.

Just trying to wrap my head around this. Any additional insight is appreciated.