Now my problem as if the above wasn't enough

I'm trying to take out my old wall cabinets and two things. The installers stripped some of the screw heads and they ones they didn't strip they filled with Ackpunky that is like old old gum. You can dig and dig but it won't come out and if you try to back out the screws without the hole being perfectly clean you will strip out the screw. Don't ask.
Short of destroying the cabinets ( have my eye on them for my garage) any one got any idea of how to get them out? I can't get to the screws with a recip saw or I would just cut the buggers. I thought of a 3/4" hole saw to cut around the screw but I don't think a hole saw without the center drill would cut. I thought about drilling them out but there is a about ten or more and I'm on a dead line. Must have them out by the mornign of the Sept 18 the dry wallers are coming to fix the ceiling.
I tried the Sears guaranteed screw remover and that is a work in frustration. Any ideas?
They are healthy screws #10's or possibly 12's
Anyone need entertainment? I can use some help.