What smarty cats you have. I have four and none of them have figured out how to open drawers. One figured how to open a sliding door by bouncing on the handle from the dresser sitting next to it.
My fabric is stored on open shelves and plastic drawers. Some even hang on pant hangers.
My kitties do like to lay on everything from cutting table to ironing board to computer chair. But most of all they like quilts.