[SoOkay]

Use a screw extractor. I try not put it on a drill, you get more control if you stick it into a socket wrench.

If the extractor can't grip, you may need to drill a small pilot hole in the screw shaft.
Either way, slow and controlled will get you way better results.

 

[SoOkay]

well this just turned into a lager project. If the screws are bonded you may not be able to back them out

on one hand it may be possible to drill thru, and bolt your new clam cleat on but if I did that I'd use a backing plate to ditribute the load

or you could drill it thru, fill it in and then reinstall.

assuming you dont need to take the wood base out anyway.