The oil type spots and streaks on your clothing could be caused by several different problems. Here are some tips to eliminate and prevent these types of stains:
Using too much fabric softener will result in oily spots on clothing. Also, adding the softener directly to the tub instead of the dispenser will result in this problem. Fabric softener can build up in the dispenser hose to the tub as well. I recommend that you clean your dispenser drawer and wipe out deposits that may have built up in the housing. There may also be a buildup of deposits in the hose from the dispenser to the tub. You may be able to clear these deposits by running an empty cycle with just bleach being dispensed through the detergent dispenser.
You should be using HE (High Efficiency) detergent. Failure to use this type of detergent can cause deposits in the dispenser as noted above. It can also build up deposits in the washer tub. A new product named AFFRESH can be used to eliminate deposits in the washer tub. If you cannot find this product locally, you can order it through Sears Parts by calling 1-800-252-1698. You can also order the Affresh online by clicking the Parts link on Sears.com. The part number for this product is W10135699.
The spots may be coming from the door boot seal. Certain conditions can cause this boot to deteriorate. It can cause the oily spots and streaks that you describe. Try rubbing a moist paper towel over the entire door boot seal. If you get residue on the paper towel, you should try to clean the entire boot. Try using diluted white vinegar to wipe down the entire door seal. Rinse it with plain tap water. Run an empty cycle with one cup of chlorine bleach added to the wash tub (no other detergents added). After the cycle, leave the washer door slightly ajar and allow the boot to dry. Test for residue again. If you are still getting residue from the door boot, you may need to replace it.
Overloading the washer can cause clothes to be forced up against the door boot and streaks can result on garments.
Certain plants or chemicals can cause deposits on clothes when laundered. Wisteria sap can cause brown spots on clothes. The sap will appear to be clear until the garment is washed. The same result will occur with certain types of weed killers and other chemicals.
This washer has a belt drive system that turns the wash basket. There is not a source of grease or oil that can leak into the washer tub from this type of belt drive system. It is not likely that your oil spots are caused by mechanical lubricants in the washer.
The above tips should help you find and eliminate your problem. If you continue to have this problem and cannot find the cause of the spots, you will probably need a service technician to diagnose and repair your washer.
The procedure for accessing the water seal and bearing in the outer tub at the back of the washer is a complicated process. It involves the disassembly of the cabinet and the tear down of many internal parts of the washer. To see the parts breakdown, you can click on the Parts link on Sears.com. The parts diagrams will illustrate the drive system and the tub assembly. The bearing that you are seeking to access is on the rear shell. To get to it, you must disassemble the outer cabinet, remove the drive belt, drive motor and related components, remove the shocks and suspension springs, remove the 23 screws that hold the 2 halves of the outer shell together and remove the inner basket. Reassembly will require the replacement of the gasket between the two tubs. It also requires precise tightening of the 23 screws that hold the 2 outer shell parts together. Failure to properly tighten the screws will result in leaks. I recommend that you contact a service technician if you want to check this bearing as a source of your oil spots. If you are completely confident that you can safely complete the complicated procedure that I described above, resubmit your question for help in accessing the shell bearing.