Go to your local DMV website and look up their statutes on changing the name on your license. Some States have a time period built into it like (for example) 60 days. So before the 60 days are up you would need to hit the social security first and then the DMV. Now I have never heard of anyone who has waited overly long before getting docs changed over so I have no idea if the local .gov might get upset and prosecute someone who does not follow the rules.
I advise against it because it can be difficult to have the dual identity like that. Trust me, I know. Find out how long it takes and then within a couple of weeks of going FT, go and get it done.