The answer to your question is: You are not too clumsy for stockings, you are just inexperienced.
First and foremost
You must learn not to "grab a handful", keep your fingers straight and thumb parallel to the fingers.Follow the good advice on the nails.
If your hands are near female norms, a pair of cotton lingerie gloves really helps reduce nail snags.
An inexpensive pair of white cotton gloves is a good alternative. Look for Disposable White Cotton Gloves, Cotton Glove liners, Cotton inspection gloves, etc.
It takes practice and care.
Find the correct size for you, buy a discount multi-pack of low cost stockings.
While imagining they are $20 a pair and they are your favorite ones, gently roll them on.
Practice, practice, practice
Soon you be so good at putting them on, that you will worry about removing them safely

Then buy better pairs.
Skip the extra sheer and go for the everyday or "energizing" ones that are stronger.
.... I still carry a spare pair in my purse, in case of unexpected fashion malfunction after using the restroom
-Cindi