Don't start with a drill, move slow with a few hand tools and the gun firmly held in a vise. I've removed many screws, and only used a drill on one of the first few, and that made a mess out of that time !
A hard, very sharp scribe is what I use. No hammer is usually needed and can actually make things worse. Place the point where it might dig in best, like a jagged area in the screw surface, at about a 45 degree angle. Lightly tap it, see if it moves, if it does, tap again and repeat going around counter clockwise. Don't smack it and don't hurry. In place of a hammer I would start with something like the side of an open end wrench, a 5/8 inch wrench is about the right heft, the flat surface of the wrench helps because you don't have to aim as precisely as say a tiny tac hammer, this way you can war to the screw.
This all might sound odd but its worked for me on screws smaller than what your dealing with up to screws as large as my finger, the force is scaled up on the larger screws obviously.
