So I guess technically Archie is currently Prince Archie of Sussex, Earl of Dumbarton as his full title?
The convention is you use your highest title, or the courtesy title you can use as an oldest son. And only one at a time, unless it’s an official proclamation or something which lists them “Queen Daenerys, Stormborn, Mother of Dragons, Queen of the Andals and the First Men, Protector of the Seven Kingdoms” style…
So, Prince Archie (a title held by Archie is his own right once Charles became King) is “higher” than Earl of Dumbarton, which actually belongs to his Dad, but which Archie could use if he had no other title. So Prince Archie is correct.
For Harry though, his Royal Dukedom ranks higher than his title as Prince. So normally he’d be HRH The Duke of Sussex. The same way HRH the Duke of Kent is almost never called Prince Edward the Duke of Kent (though he’s also Prince Edward).