I worked at a pizza place in college, & I've only seen crust get stuck in pizza screens, not onto them, & only under 2 circumstances:
1. whoever put the crust on the screen pressed it down onto the screen, so when the dough cooked, the areas that got pushed through expanded, hardened, and got stuck.
2. the screen was old & the mesh started to break & warp, & the dough wrapped around broken bits of the screen while it cooked.
So, gently lay your crust on the screen, gently spread the sauce & add the toppings, and make sure your screen is in good repair & you shouldn't have any problems. Other toppings stick to screens pretty readily though. We always just threw them back in the oven until the stuck-on food burnt to crumbs & was easy to scrape off.
Always try to bring a spare set of clothing when you dream about going to work or school. That way, if you encounter someone having that dream where they show up in their underwear, you'll be their hero.