A scale is an important investment, for exactly this reason. If the mozzarella is cold when you shred it, an ounce might fill up 1/2 cup. If it's warm, or if you don't measure it right away, the shreds start sticking together, it's less fluffy, and it might only be 1/4 cup or even less.
It's even more dramatic with things like cereal. A cup of raisin bran from the top of the box when you first open might weigh only an ounce, while a cup of the smooshed-up flakes at the bottom of the box might be 4 ounces! The smaller the pieces, the more will fit in a cup. Calories correspond to the weight of the food, not how much space it takes up. Without a scale, you could be off by hundreds of calories.
You can get a pretty decent digital scale at Bed Bath and Beyond for $25, plus they almost always have a coupon for $5 off, which brings it down to $20. It's definitely worth it, even if money is tight.