Make sure the arrow on the filter lines up with the arrow on the fridge so that the filter is properly installed, as show in this video: youtube.com/watch?v=EoYZdCpZpME
If the filter doesn't work after installing it properly and trying to flush the water as show in the video, you either have a defective filter or a bad knock-off. Some items online may appear to be the genuine Kenmore filter but are just knock-offs. If you have the genuine Kenmore filter from Sears PartsDirect, make sure the O-rings are still in place after you removed the cover on the top of the filter.
You can find info on how to flush the water properly on page 14 of your owners manual. If you have low flow with the old filter, and normal flow without any filter, a new genuine Kenmore filter should solve the flow issue once you flush the water as described on page 14.
I have the same problem. When I looked up the product manual, it said the warranty for the compressor is 10 years so it is within the warranty period. I plan on calling Sears tomorrow. Are you sure you purchased yours in 2013 because inside my ref, the manufacturing date is Aug. 2014. And I remember buying mine late 2014. Can you find out what is the cubic ft. capacity of this model. how much is the ref section and how many cu.ft is the freezer section. One thing when I looked up compressor, in the Parts Department of Sears, the compressor is no longer being manufactured and there is no alternate unit you can use. I wonder what the SEARs repair people will say. Did you get a repair tech from Sears?