I have a Kenmore washing machine that was purchased in 1988. The
agitator is making a grinding noise and partially 'slipping' as it
tries to do its thing. The suds also don't get completely rinsed away
on a normal cycle. I have the parts list and diagram for the unit,
model # 110.82892300, Kenmore Advantage. If I had to guess, it seems
like there is something wrong in the gear case, part number 389246.
It has several gears with teeth, and the sound the agitator makes
might be these teeth slipping.

I may have already answered my question, but does this seem like
something work repairing, or should I just get a new unit?

While I am posting, I had just changed the input hoses to the washing
machine, and then on the next cycle, the downstairs powder room toilet
bowl filled up with suds! The toilet is directly below the washing
machine, so I guess there might be some partial blockage below where
the two drains meet, and then when the washing machine empties, it
backs up into the toilet bowl. Not that big of a deal, and it keeps
the bowl clean! It doesn't overflow or anything, so is this something
I should call a plumber for?

Thanks!