Can a child get really dry skin about a week after a fever?

Yes. Anybody can, actually, not just children. When a fever and/or a rash resolves, it's very common to see dryness and flaking of the skin (the technical term is "desquamation") indicating that a fresh layer of skin cells is replacing the ones being cast off. No treatment is necessary, but you can apply a bland emollient lotion if you want to.