I think I first heard of it in a Saki story where the people turned out to actually be made of sugar.
EDIT: However, my personal version is “Don’t worry, I’m not the Wicked Witch of the West.” I didn’t realized that was a Thing elsewhere, though; thought I was just being a smart alec.
My husband used to say the opposite- he’d give me his umbrella or jacket, because “I was so sweet I’d melt in the rain.” He was born in the northeast but grew up in the south, but his parents were generations-back New Englanders, so who knows.
Northeast (raised in MD, parents from NJ), and I only remember hearing “You won’t melt!” if my brother or I complained about going out in the rain. Neither sugar nor witches were mentioned specifically, but it was always clearly a reference to the Wizard of Oz/Wicked Witch of the West.