I wish you better luck than I had.
Mine would skip/stall at around 8-10 min in, but play fine before and after that. I was sure it would be debris stuck on the rail, and tried many times to clean it, including moving the head to the extremes to expose all the rail, but it never changed, although every once in a while it would play through the 8-10 min spot. And the rail looked pristine and was smooth and slick to the touch everywhere.
My unit had laid unused for several years by the previous owner, and perhaps that was part of the problem. I probably never used it long enough to know if more extensive use would help it. But yes, clean up any excess oil you can find. It should not be spitting any on the CD.