Removing a vintage plastic tap head
I will try to answer this question without seeing a picture, or knowing what exactly kind of tap you have.

Typically with an old tap that is not in good condition, it would make more sense to replace the entire thing instead of trying to replace parts on it. Plumbing fixture manufacturers are notorious for constantly changing their designs and making them incompatible with their older models.

If you can see the screws, then I am assuming there is a plastic cover over the screw. If you take a thin bladed screwdriver, or a similar tool and push it between the cover and the tap, then you may be able to pry it off.

If you are unable to remove the screw cover, then you could try using a mini hack saw to cut the cover off. Alternately, you can try using a drill to get to it. Obviously, this is a destructive approach.

Once the taps are removed, then you should be able to unscrew the fixture from below and remove the entire thing. As long as you have a standard sink, then you should be able to find an inexpensive replacement.

EDIT: It is less common, but some of the covers are the screw off type instead of pry off.