I am having this issue which sounds the similar as the orginal post.
My frigidaire refrigerator (FRS26W2AWE) starting making a humming sound after a click from around the compressor. Then after about 10-20 seconds it would stop after another click. It seemed to be working fine until last night when everything quit cooling and the freezer thawed out. In looking at it this morning, it trys blowing but with the final click it stops. My question is compressor or the start up relay?