Dense pizza dough- need help! How to make lighter and more airy pizza?

I think you might not be giving your dough enough time between shaping and baking. Proofing lets little bubbles build up in your dough, letting it rise as you mentioned - stretching and shaping will deflate the dough and let the little bubbles loose, even with careful handling. Some methods of shaping will deflate the dough more than others, and if you're stretching by hand you should try to be careful not to handle too roughly, especially around the edges where you want a fluffy crust. Letting the dough rest after shaping and before baking will help the dough recover.

Also, you didn't mention your proofing method much, but it sounded like regular counter-top proofing. Several recipes I looked at prefer a cold ferment for new york style pizza, citing more complex flavors and better gluten development, and also that stretching cold fermented dough tends to produce a fluffier crust (since it holds the pockets of gas from rising through the stretching better). The pictures look quite convincing that this will let you have an airier crust. Since this would involve making your dough several days in advance, and letting it rise in your fridge, it may not be your ideal solution - but if you don't mind planning ahead (best results apparently 3-5 days), it may help.

Also, you might want to check how long you knead - a dough that isn't kneaded enough will end up dense and heavy. You might make sure your dough is kneaded enough to pass the windowpane test - stretch a marble-sized ball of dough between your fingers, it is developed enough when you can get it thin enough to be translucent enough to see your fingers through. This article talks bout the importance of kneading (and also suggests using a food processor for better gluten formation quicker), since having well developed gluten should trap the bubbles from rising better and give you a more airy crust.

Of course, getting more heat in the oven might help - it might even be all you need, and if your problems are solved after getting a pizza stone, so much the better. But these tips I found might also help you get the effect you're looking for in the meantime - or possibly stack several of them to get the pizza just the way you like it. Good Luck!

NYC Pizza (3 pies)

100% flour - 510g
62% water - 316g
2% oil - 10g
2.35% salt - 12g
2% L-DMP - 10g
1% sugar - 6g
0.39% IDY - 2g (1/2 teaspoon)

Mix 1/3 of the flour with all of the water. Rest 20 minutes. Add salt, sugar and oil and mix to combine. Add malt and yeast and mix to combine. Gradually add the remaining flour while mixing (five minutes). Rest for 20 minutes. Dump onto counter dusted with flour. Knead for up to five minutes or until dough looks smooth. Separate into thirds, form into balls, and place in (very) lightly oiled containers.

(L - DMP is optional, but is very helpful for browning when baking under 600 degrees)

You're currently making your dough with 63% hydration. For a lighter dough, you want about 75% hydration— This means that for each 100 grams of flour, you want 75 grams of water. You should be using a stone, and baking it as hot as your oven can go. 450º F will work, but if your oven has a broiler function, you might consider heating the oven up to 450º on "bake", then switching to "broil" when you put your pizza in the oven. This should allow it to cook at a slightly higher temperature.