If your hands are properly sharpened and you have the confidence and tools, you can remove the damaged screws by drilling a deeper hole in them. And then, by screwing in with an extractor screwdriver, carefully unscrew them outward. The method is relatively complicated, but it works even with deep-seated screws.
Rubber tape does a good job of providing grip. Place a wide enough rubber band between the screwdriver and the screw. Ideally, the screwdriver should be slightly larger than the one previously used to unscrew this screw. The grip is usually enough to loosen the screw, if not remove it permanently.
Get a set of extractors like this for all snapped or rusty screws. Costs pennies, and helps in the most hopeless moment. Good thing to give as a gift to your friends, they will never guess to buy it.
It is necessary to saw over the head with a bolt or dremel so that a new reliable groove appeared in it. Further you can handle with an ordinary flat screwdriver.
And no one has canceled the old-fashioned method: use pliers to pinch the screw head and twist as much as you can. Only you need to keep in mind that the metal ends of the pliers can damage the surface you are working with. If it is valuable furniture or some other beautiful panel, you need to be extremely vigilant.