I have not had any problems using any of the Spellbinders dies in my Bigshot. To cut out, my sandwich is: multi-purpose platform (all tabs closed), cutting pad, die, cutting pad. Once cut, I run it through again to emboss: multi-purpose platform (one tab open), cutting pad, die, 2 X silicon embossing mats, cutting pad.
Not that it should really make a difference, but which shapeabilities die are you using?
The cutting plates do start to curve after a while, but I wouldn't have expected them to be curving so quickly on a new machine. Once they start curving, this can impact the cutting, but you still get at least partial cutting, and just rotate the die and pass through again.
I hate to ask the obvious question, but is the die the right side up (I only ask this, as I have tried to cut with the die the wrong way...and not just once