That part also contains the temperature sensor as you can see the wound coil temp sensor on it. Its probably bad and I would replace it. Almost all fridges use the fridge section for temp control. That part is available on eBay pretty cheap.
Most of the time the issues with a basic fridge is that temp sensor or the compressor relay. There really isn't much to them and generally the compressors can last a long time as long as they don't leak.
Also, if you don't have central AC and your kitchen is like 80+ degrees your fridge is going to run like crazy. I have an electricity monitor on my house and have been away and set my central air to like 82 and saw it runs like twice as long.