Your status is not clear to me. Did you file for I-485 (AOS)?
If you didn't, it's not possible for you to get an EAC (as far I
know).
Even if you apply for EAC (provided your I-485 is pending), you
are not going to get it within one month.
You need to talk to your lawyer. Probably H1 extension will be
faster, but I am not sure.
Side Note: An EAC itself does not ensure the legal status. It's
only an authorization for employment. Normally an EAC is
issued based upon already existing and valid legal status (for
example I-485, J1/J2/F1 visa, political asylum etc).