Ive used a little tiny hole saw before to drill around the old screw, andthen chip out the cut wood with a tiny chisel (or old screwdriver) just enough to get some grips on it and turn the rest out. This obviously doesnt do any wood you will be reusing any favours though.
They are a kind of thin walled holesaw. I've been thinking of buying a set. I've never tried them but I've heard they are good and they remove a minimum amount of wood from around the broken screw.
I have something similar but cannot remember were I got it from! It is quite good and you need a drill which can reverse its thrust to be most efficient.
Have a look at what I posted some time ago this is how I removed screws in the early days of refitting
Mercia III.
Hope this helps. Have a look at the two photos on the Old -Salt posts just click on the links that may help. There are some other very good suggestions on there to. See here.
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does it have to come out ? if not then plug hole and drill along side for new screw.maybe at an angle so as to miss old screw.
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OK, but mind the two adjacent holes don't weaken that timber too much. I had a gaff break because the previous owner had moved the halyard fitting so that there were 4 screwholes in the same plane. Roughly the equivalent of sawing it in half & hidden under the fitting where maximum leverage was applied! /forums/images/graemlins/shocked.gif
On doing my New Year sort out I found the invoice for a Screw Extractor and a Dowel Former. They are from Turners Retreat - www.turners-retreat.co.uk - and together cost me £13.90 although that was some years ago. I have found them fairly effective but a reverse action drill is needed