Consider C-3PO’s explanation of the proper protocol for addressing Leia:
“Pleased to meet you, ma’am.” Wait. Was that right? “I mean, Senator.
Or, uh, Princess—no, Your Highness!”
A golden protocol droid shuffling up behind them said, “Either Senator
or Your Highness at first reference. Ma’am or Princess Leia is
Given that C-3PO is an expert on protocol, I think we can say that Leia remains a princess.
Interestingly, a surviving Alderaanian referred to her as Queen:
It would seem logical that, in the absence of any surviving members of the Alderaan royal family, she would become queen. Given that C-3PO does not recommend this form of address later, however, I think we can say that formally speaking she lacks some qualification of queenhood (an official coronation ceremony, for example).
I believe the planet the question refers to is Birren. Since Leia was the closest relative to the previous governor (by adoption, not by blood), she was entitled to the governorship. She chose to pass the governership to Lady Carise Sindian, however (from whom it was later revoked).