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After manually draining the tub, I tried to run it through another
cycle. It fills the tub and the knob goes through the whole cycle, but
nothing else happens... it doesn't even drain the water at the end of
the cycle. During the cycle, I do hear some humming coming from the
machine (as if the motor was trying to run maybe???).

Along with this, there is a distinct smell of burnt wires or something
burning.

Probably the drain valve not opening to let the water out. You can do some
troubleshooting at:
www.applianceaid.com
www.repairclinic.com