The query_string does support some fuzziness but only when using the ~ operator, which I think doesn't your usecase. I would add a fuzzy query then and put it in or with the existing query_string. For instance you can use a bool query and add the fuzzy query as a should clause, keeping the original query_string as a must clause.
As for your additional question about how to correct spelling mistakes: I would use fuzzy queries to automatically correct them and two subsequent requests if you want the user to select the right correction from a list (e.g. Did you mean), but your approach sounds good too.