The patient: A 2004 GE Profile Arctica refrigerator, freezer suddenly icing up and temperature rising quickly, up to 20° at time of repair. However, the refrigerator part was just fine and mostly staying at 37, which was a bit puzzling. The freezer had similarly frozen up 9 months earlier, at which time all we did was defrost the frozen over coils, and then the fridge/freezer worked fine until a couple of weeks ago, when it suddenly got frozen up again like it did 9 months ago.
After reading the information and reviews on this site, I decided to replace the most likely part failure and also the cheapest part, the temperature sensor. The install video for this sensor showed me everything I needed to know and which tools were needed for the repair. First, after removing the back wall of the freezer, we saw that the coils were solidly frozen over, so apparently no defrosting had been happening for about a week.
The BEST tool I bought for this job and what I recommend everyone get is a heat gun from the local hardware store. It made defrosting quick and easy, so I didn't need to defrost the fridge for hours like I did the last time (without the heat gun). So defrosting took about 20-30 minutes with the heat gun. Then we snipped the wires of the old temp sensor and attached the new one using the red butt connectors shown in the install video. This took maybe 10 minutes. Then we put the fridge back together, plugged it in, and before long the freezer temp slowly went down. We finished the work at 6 pm and by 8 the next morning the freezer was 0 and the fridge was 37. It's now about a week later and all is still fine. If the coils freeze up again, I plan to buy the second part which is also related to temperature and also only about $20, and installation should be pretty much the same as the temp sensor. And if THAT doesn't solve it, I'll move on to perhaps replacing the defrost heater.
I'm very pleased with AppliancePartsPros.com because I ordered the part Sunday night, and paid an extra $5 for quicker shipping, the part shipped Monday and it arrived Tuesday. Also the install videos are wonderful to have and easy to follow.