I don't have a lot of advice for you, unfortunately. First, congratulations on making this tricky cheese :-) Forming it well is an art, I think.
The main advice other people have given me is to treat it really gently (although some traditional makers go absolutely nuts with it -- I'll explain). Remember that the curd is a matrix of protein with a kind of tiny honeycomb of whey and fat trapped inside. Acidifying the cheese allows the calcium to come out, which makes the curd more flexible. However, at the same time, it opens larger holes for the whey and fat to escape. As you stretch the cheese, it opens those holes even more. The trick is to kind of stretch the cheese, letting the whey and fat escape, but then wrap it back up in the cheese. The result is a kind of layering of cheese, fat/whey, cheese, fat/whey, up through the body of the cheese. Originally (more than 150 years ago) mozzarella was made with sheep's milk, which has a very high milk solid level. Eventually they moved to water buffalo which has twice as much fat as protein. The result is that the wrapped up whey is very much like cream. Very good mozzarella literally gushes this cream when you cut it. So far machines can't make the mozzarella well enough to get that consistency and it takes years of practice to get good at it. It's probably one of the hardest cheeses to do well.
As I said, some producers go completely nuts with the curd. If you look at "Alex the French guy"'s youtube channel, he has a (mostly not very good -- though he did his best) series on making mozzarella. In it, he visits a good mozzarella producer. They get a large tub, fill it with very hot water and then with a paddle whip up the mozzarella. It's worth digging out that video and watching what they are doing. Instead of being very careful and gentle, they move very quickly and wrap the mozzarella up. I've meant to give this a try, but I haven't gotten around to it yet. I think it's probably quite difficult to do well :-)
The other video I really recommend is the Buratta video from "Food Insider". While it's buratta, not mozzarella, there are a couple of useful things to see. Watch how they evaluate the curd. It's probably the best video I've seen of it. At the very end, the guy also makes a mozzarella rose (his signature) and I think this is worth it to see how an expert handles mozzarella.
One last thing. Cow's milk mozzarella is normally called "fior di latte" or "flower of milk". I think part of this is because you get the aroma of milk in the cheese. However, my personal belief is that the forming of the cheese is creates a specific internal shape. When you cut the cheese, you should be left with layers apon layers of cheese with whey in between. It should resemble the petals of a flower. It's actually easiest to see if you tear the cheese in half with your hands, for some reason. Cutting it often squishes it. However, very good cows milk mozzarella should be able to stand up to cutting as well.
I hope that helps you aim for something, even if I didn't tell you how to get there ;-)