Now that you have confirmed that I am on the right track, I digged a little deeper into the „short“ matter. It looks like the German word schurz (here in the south west indeed pronounced shots), first part of the armour, then worker's apron, does not exist before the 14th century. This makes it rather unlikelely to be the origin of the last name. What appears more likely is the literal meaning „short“ for a short person. I found Scurz as a personal name in the early 9th century, obviously a byname. So Schorz seems to be the same as Kurz, another German surname.