Oil spots on your laundry are usually the result of a bad seal around the agitator shaft. This causes water to leave the tub and run down into the transmission. Since the transmission oil is physically lighter than water, it leaves the transmission and floats into the tub. At the same time, water stays in the transmission.
The only fix is a new transmission and seal. This is a very expensive and difficult repair, and the cost may not be worth doing on an older unit.

 

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