In Disney’s Cinderella, is her full name Princess Cinderella Charming?

Based on a rough analogy from UK tradition, Charming would be the Prince's first name, not his last/family name.

Simple ref: Prince Charles.
He is Charles Windsor, but 'Prince Charles' is how he is referred to [if not being highly formal, with HRH, Prince of Wales, Duke of Cornwall...etc] so I'd say someone made a proper boo-boo calling the son Chad Charming [though presumably they did it because Prince Charming has no known last name].

So... no.
She would be titled Princess Cinderella, named Cinderella [family name of the current Royal family, unknown as it was never written down in the original fairy tale, or alternatively her original family name]
It is generally accepted that British Royals don't really have a family name and they are rarely referred to by it. The current accepted wisdom is to use 'Windsor' if you feel you need to use one at all. Windsor itself is an invented name, chosen because Saxe-Coburg felt a bit 'German' at the time of WWI.