Kenmore refrigerator model 59562998 Compressor is running but no air is blowing into the refrigerator or freezer

Hi and welcome to FixYa. I am Kelly.
With no air moving the reason will be that the evaporator fan is not running.
Causes for the evaporator fan NOT to run:
1. Ice build up on the evaporator coils (Check to see if there is a large amount of ice on the back wall of the freezer compartment as noted by excessive frost.) Press the evaporator access cover to see if it will flex slightly. No flexing at all = ice build up.
ICE BLOCK NOTED:
a. Perform a manual defrost by unplugging the unit and directing a fan into the compartment for just over 2 hours with the freezer contents removed. (Water mess so be ready!)
b. Clean up the water mess and return the unit to service. (Check to see if evaporator fan now runs) If the fan now runs you probably have an automatic defrosting problem and most likely the ice block will form in 2 - 3 days again which over time will cause the evaporator fan to stop running once the blades are encased in ice.
2. Open winding or dry bearings of the evaporator motor.
http://www.appliancepartspros.com/part_details.aspx?part_id=3140606
This particular motor if the winding is good you can lubricate the bearings with turbine oil (Zoom Spout Oiler available at any appliance parts source and most hardwares) and bring them back to service. I actually remove the bearings from the armature and vlean the armature shaft before lubrication. Then lubrcate and reassemble.
3. Wirning problem at the evaporator fan motor electrical connector.

These are the 3 most common reasons the evaporator fan does not run. The evaporator fan should be running anytime the compressor is running and the freezer door is closed. I did not check to see if your evaporator fan motor power is controlled by the freezer door / light switch. Just check to see if the light goes out when the door plunger is depressed.

Thanks for choosing FixYa,
Kelly

The evaporator fan has stopped. it is usually in the back of the freezing compartment behind all the plastic. when the fan stops no air can circulate over the cooling coils so even though the compressor is running no cold can move iside the freezer or fridge. be extra careful when you take the small screws out as the screw heads can be full of build up that makes it hard to get a good hold of them. take your time and clean off all the screw heads so they don't strip out. The evaporator fan is not to expensive wait to buy on until you get the old one out.