I imagine folding a paper in the order that they do, and then go through it backwards when I imagine unfolding it putting imaginary holes where they should be. It helps me to imagine little black circles in the dotted lines of the paper folds. I don't know if there are other ways to do this (I'm probably sounding like an idiot for explaining how I do it too and I'm sure this is exactly how most people do it), but that's how I do it.
Maybe actually do what you said, practice using papers and punching the hole yourself but don't unfold it. Try to draw it first, or open one fold if you can't guess it, then draw after having open one fold. Most people might not actually fold a piece of paper, punch a hole, then draw their answer on a different piece of paper. But if it works for you, who cares? Folding the paper yourself first might get you to start to see everything in your head. Then you can come back and let everyone know that you got a 27 on the PAT section.