Because I like consistency and reproducibility of results, I think the last piece of pizza-related equipment I would give up would be my digital scale. However, it is possible to work with volume measurements. But the only way I personally would do it would be to use forum member November’s mass-volume conversion tool at http://foodsim.unclesalmon.com/
(on the left-hand side). With that tool, you can convert weight measurements for flour and water in recipes to volume measurements. However, when measuring out the volumes of flour, you have to follow a precise set of steps. You start by stirring the flour in the flour container to loosen the flour. Then you use a tablespoon or scoop to lift flour from the flour container to your measuring cup(s) and measuring spoons just to the point of overfilling. You then level off the measuring cup(s) and measuring spoons with a flat edge of a knife or something similar. With November’s tool, you can select the flour choice from the pulldown menu. When measuring out water, you should view the cup markings at eye level. Good digital scales can now be found quite inexpensively. If you need a few ideas, I can lead you to a thread or two that discuss scales.
You didn’t indicate how you make and manage your dough but if you are experiencing snapback with your dough, there can be a few possible causes. I would say that the most common cause is underfermentation, that is, not letting the dough ferment long enough before using. A second possible cause—a very common one--is re-kneading or re-forming the dough just before shaping and stretching it. That’s a no-no since it rearranges and reorients the gluten structure in a way that the dough becomes very difficult to stretch out without its snapping back. This problem can sometimes be resolved by letting the dough rest a few hours and then trying to form it into a skin again. In my experience, it is better to avoid the problem by just not re-working the dough to begin with. Another possibility is that the dough was overkneaded, but since you don’t have a mixer it would be difficult to do that using hand kneading. If you are using a high-gluten flour, that can also sometimes lead to an overly elastic dough because of the higher gluten content of that flour. Professional pizza operators use additives to combat this problem, however, if a high-gluten dough is given sufficient fermentation and you avoid the other problems mentioned above, you should be fine using the high-gluten flour.
For a beginning-to-end discussion of how to make a NY style dough/pizza you may want to take a look at this thread: http://www.pizzamaking.com/forum/index.php/topic,2223.msg19503.html#msg19503.
The bulk of the discussion takes place at Reply 8 and thereafter but you may want to read the posts preceding Reply 8 for background purposes. You will note that the discussion at the above thread is with respect to the use of a stand mixer. However, the overall discussion should still be useful even if you use only hand kneading. If you happen to have an electric hand mixer, it is possible to use that instead of a stand mixer, along with hand kneading. Such an approach is discussed here: http://www.pizzamaking.com/forum/index.php/topic,3985.msg36489.html#msg36489
There is also a lot written about one of the more popular NY style pizzas at the so-called “Lehmann” thread at http://www.pizzamaking.com/forum/index.php/topic,576.0.html.
Since that thread covers a large number of possible variations of the Lehmann NY style dough, you can take a look at the Lehmann “Roadmap” at http://www.pizzamaking.com/forum/index.php/topic,1453.msg13193.html#msg13193
to narrow down the choices. You will note from the Roadmap that there are a few versions that rely on hand kneading. if you have a food processor or a bread maker, there are also recipes that are based on using those appliances.
While the Lehman NY style is a popular one, there are many other NY styles covered on the forum. If you have a particular version that interests you, we should be able to steer you toward a possible recipe.