Darren,
I guess I can chime in with some memories of the 1963 Fleer baseball. I collected them as a kid. The ginger cookie was not oily, as I recall, and perhaps a bit stale. Still, cookies have oil as an ingredient, and the Fleer company did something very thoughtful to prevent problems to their cards.
I remember this very well.
Their nickel-a-piece wax packs included another plain card, dark brown on one side and white on the other. The dark brown side faced the cookie, which likely was included to prevent any oil residue on the cookie from being absorbed into one of their beautiful cards. I never remembered any of my Fleers being stained. That would have annoyed me even as a 9-year-old, and I'm rather certain I would have recalled if the cards came out of the pack stained. Some things you'll remember from your childhood--52 years later!
Hope this helps at least eliminate the free cookie from being the cause of the stains on your Gibson.
--Brian Powell