No, your Home of Record is the state where you enlisted (Florida) until you intend to abandon it. Just because you lived with your mother and she moved to South Carolina does not mean that you must "move" with her. You should not have paid any taxes to South Carolina unless you had income from South Carolina (like a rental property). Also, many military use a relative's address as their mailing address when they are deployed- as long as it is reasonable that you will be able to retrieve the mail if needed.
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