Are they deposits over the sink surface or stains in the sink?
Is this a granite sink or is it an engineered stone sink?
You can clean off your oil deposits and sealers with an alkaline solution. Caustic soda works great for this (or Drano, but it is much more expensive) and you can get it very inexpensively from a pool supply store. Keep the stone moist with this allowing it to penetrate the stone. You might try covering it with paper towels after the application to keep the solution on the stone. This solution will dissolve waxes, oils, even sealers!
Then I would try the Pectro to stain it to one consistent shade.
http://www.defusco.com/tenax-pectro-black-stone-treatment-liter-p-2303.htmlhttp://www.defusco.com/tenax-pectro-clear-stone-treatment-liter-p-2477.htmlMark