Drill the heads of the screws, then pop the cart open. This way you'll save the cart from destruction, if you're careful, and you'll still gain access to the guts. You should be able to twist what remains of the screws out with a small pliers or vice grip, then reassemble with similarly sized screws.
As far as cleaning the contacts, as I've said in other threads, I use steel wool. 000 for light cleaning, 00 for really nasty buildup (which sounds like your issue).
Personally, I'd throw that cart away. I don't even what to know what kind of bacteria/mold/whatever are growing inside something that barfs out black crusty bits and whatnot
