Oily spots on clothing after wash?
You will want to make sure that you pull the agitator and check for fabric softner build up that can cause the oily spotting on clothes. If this checks ok then you will want to make sure that the shaft seal is not leaking.
Should the agitator shaft have any sideways play in it?
Answer Hello Bob. No, there should be no play in the transmission 3360629. However, there is a little between the transmission shaft and the drive tube.