Sounds like the drain pump is going bad or it could be the relay that controls the pump but, being that tapping restarts the pump, I would suspect it to be the pump.
I believe that the cost of a new pump is very close to the cost of a whole new dishwasher. Call Sears and see if they can give you a price on a new pump. Kenmore is made by GE so I would think they could cross reference a Kenmore pump.
If you decide to replace the pump, turn the power off to the unit, disconnect the electrical, water line and drain line. Unscrew the unit from the bottom of the counter top, possibly lower the leveling feet and slide the unit out of the opening. From there, you'll have to study which screws, nuts&bolts require removal to get the pump out of the unit to replace the motor. Maybe just some light oil to the bearings will buy you some time. But before you remove the pump, check to be sure there is nothing obstructing the spinning of the pump in the well inside the unit. You will have to remove the strainer basket in the inside bottom of the unit to see what if anything is in there.