Kenmore 80 series washer does everything but spin?

My Kenmore 80 series top-loading washing machine stopped spinning and draining. It performed all other functions normally. When starting the machine on spin, the agitator jerked once like it wanted to start and the motor sounded like it was trying to make it spin. However, it just wouldn't work.

We shook the tube to drain the water. Now it spins, but it sounds AWFUL.. like the motor is NOT happy. We're afraid to use it. It's only about a year old.

Any suggestions?