The title character of May is a similar example, minus the telekinesis. Poor May's only comfortable social interactions are with the doll she calls her best friend. She's so shy that she can barely talk in the presence of other people at all, even when they're clearly interested in getting to know her. Although once she's decided to kill them all, she becomes much more confident.
Funnily enough the actress that portrays May (Angela Bettis) also played Carrie in the NBC TV remake in 2002.
The protagonist in the movie Coyote Ugly is one of these. As a bonus, her name is actually Violet.
Cadet Hooks from the first Police Academy movie spends almost the entire film unable to look other characters in the eye and barely speaking above a quiet whisper, until very near the end when arresting a suspect when she screams "DON'T MOVE, DIRTBAG!". She continues to do that in every other Police Academy movie she's in. (Really, if someone that quiet started yelling like that and had a gun, wouldn't you feel like doing what they said?)
Need Shrinking Violet personified? Well, look no further than the appropriately-named Violet Parr from The Incredibles. Not only is she incredibly shy, but she also has the hair, the mumble, and a desire to be just another girl, and she can't even look the school heartthrob, Tony Rydiger, in the eye. Her family are superheroes in hiding, and her powers are invisibility and forcefields—Word of God is that these are symbolic of her personality, respectively being her wish to hide from everyone else and her need for insulation from the rest of the world. Needless to say, she embraces her abnormalities, gets an Adrenaline Makeover, and puts her hair back in the process of helping her family save the world.
And to top it all off, by the end of the movie, it's Tony who can't look Violet in the eye, instead of the reverse. Think about it.
She also discovers the ability to levitate in the center of her force field when forced to save her brother. Both seem equally surprised.
Prince Albert (later, King George VI) of The King's Speech has a painful stammer, which has left him deathly afraid of public speaking and large crowds in general. The crux of the film features him attempting to rectify this in the face of the looming threat of World War II.
Mui, the female lead of Shaolin Soccer. Hair in the face, mumbling, acne, withdrawn, eventually pulling out of life in public to become a nun. Until the final showdown.