You are not quite right.
You aren't far off with "dowdy". It means lack of style, but is more to do with being old-fashioned, perhaps, than lack of taste. Lack of taste alone could mean skirts that are too short, and colours that are too bright: dowdy is about drab colours, and could not apply to revealing clothes.
"Prim and proper" is not to do with arrogance or self-importance though. A person who is prim and proper is always polite, always follows the rules - they would never even think of breaking them. There's a definition of it on this very site you could have searched for: Prim and proper - Idiom Definition - UsingEnglish.com
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