Freddy Eynsford-Hill from My Fair Lady. He's an aristocrat, but he couldn't help sniggering when he encounters Eliza Doolittle whose "new small talk" he finds absolutely irresistible.
Evelyn from Anything Goes. It's his attempts at using American idioms that really seal him as adorable dork.
Malvolio from Twelfth Night. Poor guy gets tricked into thinking that Olivia loves him. He's such a dork when he wears the yellow stockings. He's that guy who enforces the rules without bias. He's so adorable when he's daydreaming. Malvolio's such a dork and he's given false hope by Maria, Sir Toby and a few others.
College-age Allison from the musical adaptation of Fun Home is a cute dork with her sing-song rambling and adorable behaviour around her roommate, Joan.