The likelihood of saving all the screws from damage sounds slight. At some point you gotta take the bull by the horns and wrassle it into submission. IF they break off, carefully punch the center of each remaining screw/bolt and drill it out with a cobalt bit from your local source. If you don't punch the centers you may have trouble getting your bit centered. Alternatively, use a 1/16" cobalt bit to start the center of the drilling.

Best of luck.