If the freon enter the coils at the bottom then it could be a low freon problem. What is happening is that the amount of freon in the sealed system gets sued up before it can cool the entire coil.

Here is a good site
http://www.applianceaid.com/refrigerator.html
Check out #20 and the example pictures part way down the page, they show low freon.

[COLOR="DarkRed"]You stated earlier that frosted coils indicates a sealed-system problem. Does that refer to the freon system, and if so, is that something only a professional can accomplish, and more importantly, is it expensive?[/COLOR]
Partially frosted coils can be a sealed system problem.
Yes the freon is part of the sealed system. Low freon means there is a leak somewhere in the system.
Only a pro can fix this also there is no guarantee that they can find the leak and yes it is expensive. Often it makes more sense to put the dollars towards a new unit.
That is why it is so important to diagnose it correctly.

I would force a defrost cycle to get rid of all the frost/ice. The defrost timer is in the control console. There is usually a hole in the console to let you turn the timer cam with a screw driver forcing a defrost cycle. When the compressor and fans shut off then you are in defrost. Put the cover back temporarily so it defrosts correctly. Before doing this you may want to check that the defrost heater does come on. Be careful that you do not burn your fingers.

Let it run the defrost then turn the unit off and check that the coils are clear.
Put the cover back and let the unit run.
I would then check it every hour or so to see if the coils frost up evenly or not.

You are welcome
It is just too bad that the outcome was not better.

It sounds like your appliances are messing with you and you are having a bout of bad luck when it comes to them being repairable.