Any machine tool supplier can sell you LH drill bits for a couple bucks.
Buy a couple of a few different appropriate sizes to fit your needs.
CTire sells a set of screw extractors that are meant to be used in a similar manner, that look a lot like a backwards countersink. Them and LH bits require a drill that will run in reverse, without the chuck getting loose. Only ever seen one drill press that would run in reverse, which would be an ideal machine to use, if you could. A drill press can be used to hold pressure and keep alignment on a screwdriver tip in a screw while you turn the screw back, to keep it from camming out.
I have used a small hammer to tap a driver tip in to the mangled remains of a screw on many occasions. Drive the tip in, then put the adapter on it carefully, so as to not move the tip from the screw, apply lots of pressure to keep the tip in the screw, and then slow torque to undo. Worked a lot of the time, and saved drilling the heads off or having to eze-out the screw.
Have also chopped down the ends of part worn driver tips to get to an unworn or lesser worn section, before driving that into the screw.
Yep, even resorted to finding the closest 'too big' screwdriver tip and driving it in to the screw head in some cases.
Cheers
Trev