If you enter the US on a J1 or J2, the I94 form typically says: D/S (duration/status). That means, as long as you are holding a valid DS2019, you are in valid status.
I don't believe that you would have to do anything to change from J2-->J1, getting your 'own' DS2019 would be enough. In other cases, you could file a 'change of status' request with USCIS while you remain in the country.
(I don't ask my attorney for health advice, before you take my word on this please confirm it with a professional)