Hi there. The compressor start relay is indeed the most likely cause of your problems. Frigidaire was kind enough to locate a sticker on the back of many fridge models which identified the wiring diagram and the model number of the relay.
If you are not a "stickler for 100% originality" you can install a newer style of relay which will take the place of that one. There are some small differences in the internal works of the modern relay versus the original. Someone familiar with these fridges and with an "acute ear" can hear a difference in the compressor startup sound. If you're not concerned about a small change in the startup sound, I would simply install a modern relay.
The Supco RO81 or RO41 are commonly used. Some of the guys here may be able to tell you which one would be most suitable. The RO41 is for larger sized compressors which require more running amps. If you look at the compressor tag and see the running current or horsepower rating, that will tell you as well, which to use.
Often, over time, the contact points wear away in the original relay, preventing it from making contact and engaging the start winding of the compressor motor. The result is the sound and symptom you're having.
The second and much less likely scenario is that the compressor has suffered some type of mechanical damage and is seized or locked up. This could happen if the unit was run for years with the condenser clogged with dust, or if the condenser fan underneath the unit has stopped working.
The RO81 / 41 devices are very inexpensive so it's a good bet to try that first and see what the results are.
