If you pull the unit out away from the wall there is a cover that covers the compressor area of the machine usually made of cardboard or metal unscrew the 1/4" or 5/16" screws and remove this panel. You will then be able to see the compressor on the side of the compressor is a small black box inside this box is the start relay and overload. This is the part that you would need to replace in order for your unit to be up and running again if you click this link the parts are listed under 10 and 11 as relay capacitor and capacitor http://www.searspartsdirect.com/partsdirect/part-model/Amana-Parts/Refrigerator-Parts/Model-ABB2224DEW/1268/0161000/M0708009/00007?blt=06&prst=0&shdMod=ABB2224DEW If you have any further questions message me back and please also remember when our chat has fully concluded to rank how this opinion has helped in your appliance situation it allows me to continue helping others in similar situations in the future Thanks
Rick

It sounds like you are hearing the overload on the compressor closing. When the overload closes the relay tries to start the compressor (that might be the buzzing sound) and if the compressor doesn't start the overload opens and turns the compressor off. You will need to replace the relay on the compressor.