Whirlpool upright freezer making buzzing noise and not working

I had a similar problem yesterday. I woke up and the freezer was not working anymore. It would make a buzzing noise and then it would click and shut off. I looked online for some possible solutions. Here is what I did:
I unplugged the fridge. I pulled it out and took off the cardboard backing. I used an air compressor to blow out all of the dust and dirt that had accumulated on the condenser coils. A shop-vac just wasn't powerful enough to pull out all of the dust/dirt. After that, I plugged it back in....and the same thing was happening. It would buzz, and then it would click off. I figured out that the "click" was coming from a starter relay (a small plastic piece attached to the big black compressor). I detached the plug from the relay and then detached the relay from the compressor. The relay was burned out. It turns out this is a common problem with the relay. I replaced the relay for $50 (from an appliance parts store) and put the whole thing back together in about 10 minutes.
I plugged the fridge back in and the compressor started, and now the freezer works better than other. I saved about $400 doing it myself, all in about one hour. And better yet, my wife was very impressed. Not bad for a doctor.

Compressor is not starting. Could just be the relay or a failed compressor. Without a compressor starter to test it would be a guess but the relay has been known to fail.
Eric