I would try the alcohol. I've used it before. Dip the qtip, and then roll it on a paper towel to pick up any excess. You may have to rub pretty hard to remove stickies on the surface, but as long as it's not overly wet and dripping, you shouldn't have any problems.
Remember in Toy Story 2 when the restorationist washes Woody's eyes? That's what this reminds me of.