wrote

Our GE washer is about 20 years old, and the faithful ol' thing
has developed its first serious problem. It fills with water
fine, but when I pull the knob to begin the agitation cycle it
does not agitate. Instead it makes a noise that's a bit grindy.

Coincidentally, I noticed that the water pump leaks. That
couldn't relate to the above problem, could it?

There is a spline on the agitate shaft and one inside the plastic agitator.
One or both are most likely stripped out. This problem along with a leaking
pump and the age of the machine call for a new washer.

Whirlpool offers a Super Capacity washer for under $300. I think this is one
of the best buys out there.