This was my first attempt, ever, to repair a washing machine.
After some internet research on the problems, I was able to determine that the shaft seal on the gearcase probably was leaking water into the gearcase with each use. No water was ever found on the floor, but when I turned the washer over to inspect the gearcase, I found a black, oily residue on the outside of it along with the same residue on the pulleys, the belt and the motor. I ordered a new gearcase, shift actuator, splutch kit and belt on-line with a total cost of right at $250 (much less expensive than local appliance repair facilities and certainly less than a replacement washer).
My main concern was with the removal of the tub seal. Fortunately, on this model, no special tools are required. Once the original gear case was removed, I was able to push the seal out with thumb pressure. The new gear case came with the seal already on the shaft. The seal seats properly when the gear case mounting screws are installed.
Because the gearcase is somewhat heavy and is awkward to pull once the four mounting bolts are removed, I recommend removing the motor and all pieces attached to the gear case before removing it from the washer. All of the parts can be reinstalled once the gear case is in place and secured to the washer. Of course, clean the oily residue off of everything before reassembling. Also, the Sears replacement parts website has the best video for replacing the gearcase on this particular washer.
After a couple of weeks, and approximately 15 washer uses, everything is working perfectly.
One other suggestion: The tub in which clothes sit when they are washed should be inspected after it is removed. I found over half of the holes around the outside wall to be clogged. Take the time to clean the residue with a stiff brush and some dishwashing detergent. The new parts and those unclogged holes will make the washer do what it is supposed to do: get your clothes clean efficiently.