Not necessarily but I'm not sure why that would be important. They are separate airlines until Single Operating Certificate is granted. Even after that, my guess is that until the websites are merged, there will still be US Airways and American.

Judging from past mergers, obtaining a SOC and eliminitating the other brand didn't happen right away:

Delta/Northwest - December 2009/January 2010
United/Continental - November 2011/March 2012
Southwest/AirTran - March 2012/December 2014

Thus, I wouldn't count on everything being AA in January 2016. Everything has been going at a very slow pace, so I wouldn't hold my breath. Even within 30 days of FF programs merge, they still can't pin point a date.

But all of that is irrelevant because regardless of who operates the flight, i. e. AA or US, I'd be more concerned about the type of aircraft servicing the route. I haven't seen any announcements as to what they intend to do to bring the fleet together - is MCE staying? If so, when will it be added to US fleet? IFE - AA has it, US doesn't. What will stick around going forward? I'd be more concerned about that than who operates the flight. Just because eventually everything will be operated by AA, doesn't mean you'll get the AA experience on every aircraft.