I know exactly what you mean. A lot of the stuff sold in regular shops feels more like fluff than proper cotton wool, and it can be frustrating when you actually need it for careful use. I’ve had the same experience, especially with cheaper brands, they look fine at first but turn into a mess when you try to pull off a piece. What helped me was checking suppliers that specialize in bleached and hydroscopic cotton products, since they usually offer higher quality. For example, here’s one place that has cotton wool in different forms and seems to focus on proper standards
https://cottonpulp.com/products/cotton-bleached-hydroscopic-products/cotton-wool/ Maybe take a look and compare, it might give you a better option than random supermarket packs