Thanks for your good advice. The temperature in the freezer after measurement is 26F (or -3.3C).

I thought something sounded wrong as the freezer temperature is way too warm.

Here is a site with info on how to check the compressor.
How To Fix a Refrigerator - Appliance Repair Guide - ACME HOW TO.com

Before starting the testing short your meter leads together so you can see if there is any zero offset in the meter.
Also use it's most sensitive resistance scale.

Note that the testing of the compressor windings is not definitive.
In other words it can tell you if the compressor windings are bad but does not tell you that the compressor is good as it may have internal mechanical problems.

[COLOR="Blue"]I had thought the compressor stopped running and only the condenser fan was running (determined by visual) when the noise became low. But that might not be true as I could feel slight vibration of the copper tube connecting between the compressor and the condenser. Looks like the vibration came from the compressor (is it right?).[/COLOR]
The compressor will vibrate when it is running.
I would try to locate what is causing the noise re: hi/low volumes.
Could it be the evaporator/freezer fan?

[COLOR="Blue"]Do you think I should worry even though the food in the freezer is frozen and in the fridge is cool? If so, what is the cause and how it should be fixed?[/COLOR]
You did not say if the compressor and it's fan is running all the time.
If it is then you still have a problem.
It should cycle on/off as it maintains the set point temperatures in the unit.
Since the evaporator coils are now not heavily iced up this could be caused by a sealed system problem or may be that the doors are not sealing correctly.

If the unit is cycling then the unit is probably OK but I cannot say for sure without actually being there.

If the condenser/compressor fan is running then the compressor should be running as they are wired in parallel.

Check the freezer temperature. It should be between 0 and 5 degrees F.
If it is lower than that could be why the compressor is running all the time re: it is trying to reach a set point temperature but since that temperature is too low it cannot get there.
This type of problem is related to temperature control.

Remove the cover inside the freezer and check the evaporator coils.
They should have a fairly even coating of frost.
If therer is frost in only one area (usually where the freon enters the coils) then you have a sealed system problem.
This will ber an expensive repair as you need a pro.
Be sure that the compressor has run for at least an hour otherwise you may come to the wrong conclusion.

[COLOR="Blue"]Is there anything wrong with the electronic control?[/COLOR]
Yes it is a possibility.

[COLOR="Blue"]How to diagnose it or test it?[/COLOR]
Try running the diagnostics that I included as an attachment in an earlier post.
The thermistor tests are important.
You could also check them with a meter.

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Or, can I simply adjust the temperature on the control panel display to fix the problem?[/COLOR]
This is what I would do first to see if it will start regulating the temperatuer and start cycling as it normally should.
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Which one could cause the compressor constantly running?[/COLOR]
The control board is under the unit in the machine compartment.Go to the Sears parts site and look in the Unit Pars section. The board is marked as Item 19.
Sears does not share it's part breakdown diagrams with other vendors so you cannot see the breakdown in AppliancePartsPros.