AWhere would I get one big enough and sturdy enough for a full sheet cake?!

The term "full sheet cake" can mean different sizes to different people. 

I'd put the cake on a piece of 3/16" foamcore coated with melted edible soy or beeswax and cut to the exact size of the cake.  This makes for easier moving of the cake on the counter, in and out of fridge, etc.

A 20x30 1/2inch piece of foamcore will do nicely for the decorative underboard and can be cut smaller if needed.  It can be covered in fondant, food safe foil, and fabric/paper laminated with clear Contact paper for a nice presentation.