Frameless shower door slipping
the screw heads are starting to get chewed up


They are probably machine screws. I'd take one out and take it to a hardware store and buy a set with the same thread but with hex, star or square-drive heads. The length is not critical because you can always buy longer than needed and shorten them with a hacksaw (keeping a nut on the screw to straighten the thread with after cutting)


  Is there a better solution


The shower door manufacturer may be able to supply replacement parts. I'd contact their customer support for advice.

Otherwise, after supporting the door securely, I would disassemble one hinge completely to see what is inside. Maybe there's a gasket of synthetic rubber that is supposed to grip the glass. It might be worth trying to supplement or replace that, for example, with a rectangle cut from a bicycle inner tube and maybe roughened a little.