My cupcakes with oil are not greasy. I do use a quality greaseproof liner and I have used Reynold's with success. One recipe I use has a lot of oil (Cakeman's).
Could it be your pan? If the cake counterpart is not oily, see if the cupcake pan may be a cheaper pan. You may get a more stable result from a heavier/lighter color pan. Just a suggestion.
Also check your temp in your oven. A slightly off temp may not affect a cake, but is more likely to affect those things with shorter baking times. OK, I'm out of ideas.