The agitator may need to be cut out! It seems to be loose and worn out on the bottom, he said. The agitator drives near the top, not at the bottom. It's a one-piece item so there's nothing to become "loose and worn out on the bottom." More typically, the splines at the top of the agitator drive shaft are rusted, in bad condition, so the agitator slips instead of having positive drive/movement. The matching splines inside the top of the agitator also wear down.
Sears had numerous different agitator designs. This one is known as the PentaSwirl. There are others that will fit if you can't find the specific design. However, whether a replacement (of any type) will be successful depends on the condition of the splines ... and removing the existing agitator is required to examine them.
This is a 1974 Lady Kenmore, last of the keyboard/pushbutton models.
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(TX, U.S. of A.) 
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