My bet is that this is the type of machine that has brushes in the motor. They are metal brushes that get worn and eventually need replacing. Someone familiar with motors and brushes would probably be able to diagnose and make the fix; however, it could get pricey if you have to pay someone to do it.
There are vintage sewing machine groups on Yahoo, and probably one specifically for Kenmores. Someone on one of those groups could explain how to figure out if it is the brushes.
I would probably keep the machine to try to figure out how to fix it. Meanwhile, though, I would buy another machine while figuring out if this one can be fixed. Never hurts to have two machines!