I have felt and continue to feel your pain. We had this problem with a top loader. Our washer came with the condo when we bought the unit and was probably about 8yrs old when we moved in, but the oil stains only started a couple of years ago. My husband did TONS of research on the internet, and we held off on a new washer after he learned that people were continuing to have the problem after replacing their machines. We tried EVERYTHING and in a very systematic approach to rule things out. The problem is definitely in the washer and not the dryer. It happened to us regardless of the detergent and how much detergent we used; we used as little as 1oz, but then I didn't feel it cleaned the clothes well enough. We've never used detergent with fabric softener or added liquid softener during the cycle. Basically, nothing worked: vinegar in the rinse cycle, soaking in oxy clean, varying amounts of detergent, washing clothes inside out, etc. We never called for a repair because from other postings my husband read, it didn't make any difference.
We finally bought a front-loading washer. All of the tubing was replaced when it was installed, and we're still having the problem. This has been the bain of my existence of the past few years (if you can't tell). I'm convinced it has to do with the water source (we live in the city) or residue in the pipes. My husband doesn't have the problem with his clothes, so he's convinced it's the body lotion that I use and that we use on our kids (Cetaphil cream) since he doesn't use lotion. I hate to be the bearer of bad news, but if you find a solution, please let me know ASAP! Sorry this isn't much help to you, but I thought I could at least pass along our experience :(