The reason I asked about using a screen is because the dough ball produced following the Lehmann recipe you used is around 20 oz. This is at the low end of dough ball weight used by most pizza operators to make a 16-inch pizza. Most stones are not big enough to accommodate that size. If you make a smaller-diameter pizza to fit your stone size, the crust will be thicker. You can remove a piece of the dough if you'd like in the future, but you may instead want to calculate the amounts of ingredients you will need to make a dough that will almost exactly fit your stone. Maybe I have already worked up a recipe at the Lehmann thread that will work for your size of stone, so if you will let me know what size your stone is (and whether it is round or rectangular), I can look into it. The pizza screen, particularly a 16-inch screen, is a good idea because it allows you to make a 16-inch pizza and cook it until it sets up and becomes firm, at which time you can slide it onto your pizza stone for final browning. The pizza will hang over the side of your stone a bit but it will be rigid at that point. Some people like using the screen alone, but my preference to date is to use both the screen and pizza stone for the 16-inch size.
I also temperature control the water I use in my pizza dough recipes, to achieve a finished dough temperature of around 80 degrees F. Not everyone does this, but apparently member DINKS does--in order to achieve consistency from one dough to the next. With me, it has become a habit. If you are interested in knowing how to calculate the water temperature to use, you can refer to some of the postings at the Lehmann NY pizza dough thread. Also, just because the Fleishchmann yeast package says that temperatures over 120 degrees F can be used, that doesn't mean that you should do so. Many of the so-called "Rapid Rise" yeasts were developed to allow one to make a pizza in record time. In fact, if you go to the Fleischmann's website you will find many pizza recipes but just about all of them are intended to be made in about an hour or so. Fleischmann's markets differently to pizza operators than to consumers, and, according to an email exchange I had with Fleischmann's, even promotes a somewhat different strain of yeast for professional bakers.
Another point to keep in mind is that a NY style dough like the Lehmann dough normally requires about an hour or two at room temperature (after coming out of the refrigerator) before shaping. After that, the dough should be OK for about another 2 to 3 hours. In my experience, when I am making several pizzas one after the other, the last piece of dough is usually the easiest to handle.