I was also going to suggest a Xyron....you can run it through the Xyron first and then cut with the cricut. It does not gum up the blade, it cuts beautifully and it does not leave residue (in between the cuts with more detailed cuts). I do it all the time.
3M makes a paper that's sold at J's that is adhesive backed and okay to use with Cricut machines. Is that what you are looking for? It's 8.5 X 11" but for the $$ I'd use regular cs or pp and then a Xyron or a Zig 2-way or even a tape runner on larger pieces.
I agree with this. I use a Xyron X (similar to what I've linked below, but not that specific one). It works wonderfully for small letters, etc. Anything larger I can use my regular adhesives.
I'm looking for 2-sided adhesive sheets to back paper I use to make lettering and other die-cuts from my Cricut. I know the paper exists. I just don't know the brand names or stores/sites that carry it.