KitchenAid model #KSR125F. We got it after Rita in 2005, so definitely not new. It has been a good refrigerator up until now though.
Below is what I could come up with. They have Q&A's if you click on a part and scroll down...dor the issues you seem to be having the general consensus seems to be with either the defrost thermostat or the control thermostat, then possibly the control board if those check out ok. I found a how to for checking the defrost thermostat that I will post under the link...hope this will help. Of course you may find it less of a headache to just replace the fridge
My KSRI25F Model Overview from PartSelect.com
A complete guide to your KSRI25F KitchenAid Refrigerator at PartSelect. We have model diagrams, OEM parts, symptom–based repair help, instructional videos, and more
www.partselect.com
How to check thermostat:
I suggest testing the Bi-Metal Defrost Thermostat, PartSelect Number PS11750673. The best way to do this is to remove the thermostat from the refrigerator so you can control the temperature of the sensor. You can let the sensor warm up to room temperature. With the sensor warmed up to room temperature the sensor should read approximately 6.2K Ohms at 68°F. With the thermistor submerged is a glass full of ice water, the sensor should be very close to 32°F which should read approximately 16.3K Ohms. If the values that you are getting are far off from these readings, then the sensor is bad and should be replaced. You can also check the Defrost Heater, PartSelect Number PS11740459 for continuity or visible damage. If both parts are good you may need to replace the board.Hope this helps!
EDIT: I'm not sure those are the correct part numbers for your model in the how to...however that how to came up under one of the parts for yours...just double check the fitment dirst.