Rachel Beaver your 665 model number prefix means that it is made by Whirlpool. the Whirlpool error codes list F6 as a failure of the door switches. You could see if it works by gently lifting up and alternately pushing down on the door / handle. I believe the Whirlpool also has a diagnostics mode. You can enter it by:
"1.While pressing the CANCEL keypad:
a) Open the oven door.
b) Unplug the microwave oven for 2-seconds and plug it back in.
c) Release the CANCEL keypad and close the oven door. You will see the following display.
NOTE: The humidity sensor test results will appear on the display after 8-seconds.If the tests are successful, "OK " will be displayed. If one or more of the tests fail, "ERR " will be displayed.. 2.When a keypad is pressed, a number according to the test chart will be displayed. When no keypad is pressed, a blank "—" is displayed."
The sears manual refers to F6 error as a F6 —Microwave Relay Failure, which would explain why the motor is "humming" The relay is part of the PCB and may require you to get a new PCB. Again, try the diagnostics and see what it tells you. Whatever the cause may be, I strongly recommend that you get a qualified person to repair your microwave. Those things can kill you and are not really a good project for the DIY'er unless experienced with microwaves. Microwaves are very dangerous, they can still shock you even unplugged. Here is a site with great info. Hope this helps, good luck.