checked for ice on the coils.
Were the evaporator coils heavily iced/frosted up?

I unplugged the unit for about four hours to defrost any ice I couldn't see and it soaked a bath towel.
This does sound like you have a defrost problem.

Now two weeks later the ice cream is mush and the fresh food section is at 60 degrees.
Again sounds like a defrost problem.

Remove the cover so you can see the evaporator coils and determine if there is a defrost problem.
Force a defrost cycle. I beleive your defrost timer is at the top of the fresh food section. There is usually a hole that you can insert a screwdriver into and turn the defrost timer cam. Turn it till the compressor and evaporator fans turn off. You are now in defrost.
Check to see if the defrost heater comes on. Be careful you do not burn your fingers.

If it does heat the defrost timer is shot.

If not the problem could be the defrost heater or defrost thermostat.
Note that this does not totally eliminate the timer as a cause. It could be it's contacts are shot.
Check heater and thermostat for resistance.

Here are a couple good troubleshooting help sites.
http://www.applianceaid.com/refrigerator.html
Refrigerator Repair Guide: How To Fix a Refrigerator - ACME HOW TO.com

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.