Just thinking aloud here, but if her father had died then maybe George Wells was her step-father. She was possibly known by the everyday name of Elizabeth Wells, but when she had to produce her birth certificate for marriage purposes it showed her original birth name and therefore she gave her birth father's name for consistency.
For her second marriage, she would have been able to show the marriage certificate for her first marriage as proof of her (married) name and thus it was less problematic for her to put the name of George Wells (the man who actually reared her) as her father.
Possibly an explanation or possibly just a flight of fantasy, but it could explain why you haven't found Elizabeth Johnson prior to the first marriage; because she might be listed in census documents as Elizabeth Wells. I haven't made any attempt to check out this theory in case you've already been down this path.
I note that one of the witnesses to the first marriage looks like "L Wells"
