You've probably solved your sticky problem by now, just thought I would add a comment for anyone else who might have the same problem. I've been searching about it after talking with my parents whose 1958
Caravanner accordion dividers are sticky as well as their later model wood grain refrigerator door. It reminded me of the vinyl sleeves of my high school letter jacket which. I graduated in 1991 and the jacket has been hanging in a closet ever since. After my brother discovered his had become horribly sticky, I went home and dug mine out and it was awful. It's something about material itself, the plasticizer in the vinyl separating out and coming to the surface rather than cooking residues or other things that need to be cleaned off.
Searching Google shows that the same problem apparently plaques the faces of old barbies and other dolls and they recommend a couple of plastic cleaners such as this one for this specific vinyl problem:
and another called 9-1-1 or something
I haven't tried them yet, just thought I would pass it along