Welcome MBoardMan:
The corian countertops I have seen were stand-alone; didn't need a sub surface. I have also seen lights under a corian top which gives a nice effect. Yes, it can be glued directly to the cabinets. I would get rid of the old tops, especially if it is the roll-formed type, they would interfere with the glue in that you can't get full bearing.
I agree with Jessie on the PL products to glue with.
We like pictures too; they help us to give better advice and to celebrate a job completed with you.
Glenn
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