I'm guessing you could find some plastic feet that you could replace those sticky feet with. I'd wander over to Home Depot, and stand around the hinge and knob section until one of the Home Depot people spots your blank stare and ask you what you need.
Another thought would be Gorilla glue; the stuff expands to fill voids and is tough enough for what you're up to.
Or you could coat the existing sticky feet with something that would make them ............. unsticky. The wife has this hand lotion that costs somewhere north of $100 an ounce that is impossible to get off once on your skin, and is slippery as a wet eel. She thought I should use it for the alligator skin on my feet, but forgot to consider the consequences.
Thankfully, we had good medical insurance at the time and you can hardly see the dent in the back of my head anymore.
Oh, and the stuff is not very effective on alligator skin. But it is slippery.