10/15/12 UPDATE: Picked up the D600 from Bel-Air Camera today. Their repair guys said they found no oil on the sensor, just dust. They said they were able to clean it with just a blower. I was surprised that the spots were just dust given all the characteristics I saw, and they said you can't really tell unless you look at the sensor with a loupe.
I took some test shots at f/8 through f/22. The spots that I saw were gone (there were a few specks that were visible at f/22 if I maximized contrast). See the test shots below. So it appears I didn't have any problems with my D600 after all. Meanwhile, one of the staff showed me how to use a sensor cleaning kit, which I will try on the D70 and I'll post about it then.