The indices $\alpha,\beta$ above aren't even correctly contracted. And again, no, "fermionic string" is an obsolete name for type I/II D=10 string that was used up to the mid 1980s and it contains both bosons and fermions. Moreover, you're using "Dirac action" in a strange way - the Dirac action is primarily the action for any spin-1/2 field, especially in d=4.