[Stew Masche]

This happened to me on a cabinet knob once. A friend of mine, who does woodworking, had a small tool similar to a miniature hole saw, and he said they were made for this very purpose. It cut a small core out of the wood, with the broken screw at the center, and extracted the whole piece. He then had a correct size dowel that he glued into the hole.
After the glue dried, he drilled a pilot hole for the new screw, and the knob covered the plugged area. I don't know what they are called, but it worked like a charm. If there is a store that sells wood working supplies near you, possibly they could help.
Crosscuts idea is along the same line, and may be faster than looking for this odd tool.