There are two types of screw extractors.
There are these where you hope you can get a bite of the broken screw and back it out.
http://www.rockler.com/5-piece-damaged-screw-extractor-set-with-case
And there are the barrel type that leave that hole you talked about.
http://www.rockler.com/screw-extractor
Personally, I would use the barrel type and fill the hole with a piece of dowel rod and touch up the paint.