Congratulations. I think you are headed in the right suggestion.
I'd like to suggest that you consider an overnight or 2 to 3 day cold fermentation for one of your future efforts. You might have to make a few changes to your recipe but they would be relatively minor. I estimate that your recipe makes around 33 ounces of dough. That would be enough to make two pizzas about 14 inches in diameter. From one of your photos that seems about what you made. Is that correct? I think you will see improvement in the crust texture and flavor with the cold fermentation.
I wouldn't give up so easily on using your stone, but you should by all means try using a pizza screen. I very often use both a screen and the stone. I start with the screen and then transfer the pizza to the stone after several minutes. I also frequently use the broiler toward the end.
I'd also like to suggest that you try another recipe. It is the one set forth in Reply #24 at the Raquel thread, at http://www.pizzamaking.com/forum/index.php/topic,1258.20.html
. I mention this recipe in particular because it produces one of the best handling doughs I have ever made, and you indicated that you like to toss doughs. This is the dough to do it with. I made a pizza with the Raquel dough recipe tonight and it was excellent. I used the screen/stone/broiler method mentioned above. I should be posting my results over the next day or so on the Raquel thread.