A fan is running but no cooling on either side. Before replacing it, I'd like to determine that a relatively inexpensive fix is not possible. What might I check to determine the cause?

Thank you, denman.
The compressor is not running. The only visible fan (#14 & 15 in the diagram) is mounted on a bracket attached to and blowing through the condenser (#9 in the diagram). I see no other fan. Nothing is frosted over. The entire refrigerator unit (both sides) is at room temperature when plugged in and the fan running.
If the compressor is defective, I am likely to spend the $1,300 (ouch) or so for a new unit rather than the cost of parts and service for a 10+year old unit. My hope is that something like the 'overload/start relay problem' is the cause.
Plugged into the side of the compressor with 3 round pins is a white plastic 2"x2" box with 4 removable wires attached: white, blue and a double orange. It appears to be the overload/relay shown as #4 in the diagram. The white and one of the orange wires run to a smaller 1"x2" black box that appears to be the capacitor, #6 in the diagram, bolted nearby to the side of the refrigerator. Could either of these be the culprit preventing the compressor from starting? If so, can they be tested before investing $55 or $17+?
Thank you again for your earlier timely response. I look forward to your additional thoughts.

denman, The compressor start relay #SP1034 is cooked inside. Should replacing it solve the problem? Other thoughts?

Yes replacing it should solve the problem.
Part Details - WHIRLPOOL OVERLOAD/RELAY KIT, part number: AP4009244

I would check the compressor windings following are a couple sites which show you how to do this. Note: remember to also check them to the case.
I would also check the capacitor just to be sure it is not a dead short.

http://www.applianceaid.com/frig.html
Refrigerator Repair Guide: How To Fix a Refrigerator - ACME HOW TO.com

I have a CEN-TECH digital multimeter with 8 separate ranges but am not sufficiently competent with it for these tests.
Note that I am not familiar with this meter so the below is general info.

1. In order to discharge the capacitor, after unplugging the refrigerator, can I short the 2 terminals together with an insulated screwdriver?
Yes

2. On which of the 8 options do I set the meter to test the capacitor for a dead short? Then, how do I interpret the resulting reading?
After shorting the capacitor leads put the meter on a mid-range ohms setting and measure it. You should see the reading increase as you read it. This is the capacitor taking a charge. If you just get a stable ohms reading or no reading try a higher ohms scale. What you are looking for is that you do not get a o hms reading (dead short). Measuring a capacitor with a meter only sort of works as it does not tell you if the capacitor will fail or leak voltage at higher voltage.

3. ‘ACME How To’ has an excellent graphic display showing compressor testing. My issue again is setting up the multimeter. ACME says:
‘Set the multitester to the ohms setting X1’
Assuming the meter should be set to the Ω range, mine has 5 options ranging from 200 – 2000k. Where do I set it since there is no X1?
ACME doesn’t mention: ‘remember to also check them to the case’. How is this done?
Note that X1 is usually for analogue meters.
I would use the 200 setting. Your meter is showing the dynamic range.
In other words on 200 you can read resistance from 0 to 200 ohms.
Anything above 200 shows as an open.
On 2000K you can read 0 to 2,000,000 ohms. But lets say you want to read a resistance that is 190 ohms, your meter would show this as 0 ohms because it does not have enough digital display digits to show 190 ohms.

4. The meter also has 3 port options for the probes.
10ADC, VΩmA or COM? Which port gets the black probe, which gets the red?
COM is always black lead
VΩmA is for the red lead:
Voltage, you probably have a button to switch to AC or DC AND
Resistance (Ω) AND
Low current milli-amps
10ADC is for the red lead when you are reading amperage up to 10 amps

Things I do:
I always touch the two leads together when I read resistance, this tell me that the meter is working and also if measuring a low resistance if there is a 0 offset
If I measure a resistance component and get an open (infinite resistance), I always try higher scales in case it is out side the dynamic range of the meter setting.
If I measure a resistance and get a short (0 ohms) I always try a more sensitive setting so I can see if there some resistance there or is it a true short (0 ohms)

Thank you for your patience.
Always happy to help someone who is trying to learn.

IT'S COOLING denman!!
The modified relay kit was a challenge to install for a neophyte but apparently I did it correctly. Fingers crossed that Ms. Jenn-Air will still be happy tomorrow and beyond.
THANK YOU for your outstanding support! You get a gold star!
Bob