Victoria stopped using any reference to Alexandrina when she was about 7 when she stopped being called Drina in the family. The Court Circular refers to Victoria as Princess Victoria during the reign of William IV which would tell the world that is the name that she used. It was the name by which the public knew her.
e.g. Court Circular 23rd May 1837 - The King, we are happy to state, is better. The State Ball, at St James's Palace tomorrow (the anniversary of the Princess Victoria's birthday), will, it is expected take place.
Court Circular 25th May, 1837 - Yesterday, being the birthday of Her Royal Highness the Princess Victoria, Kensington was more than usually gay.
No mention of Alexandrina in the CC for Victoria.
In addition - the articles in The Times about Victoria's 18th birthday all referred to her as simply Princess Victoria - no Alexandrina.
For anyone wondering where I am getting the CC from - I am using the Archive from The Times. I subscribed specifically to get this information and am slowly ... very slowly spending my retirement (between bouts of pneumonia and two knee replacements) doing an analysis of the CC. I have started in 1952 and will work through the Queen's reign, minus 1986 - 1996 as The Times Archive stops in 1985 and the British monarchy doesn't start until 1997. I thought I had a means of getting the missing years but it seems the State Library of NSW hasn't put the entire Times on their archive page and only put up selected stories. I don't know if I will be able to physically get to the Library to see if they have the hard copies and then how I might be able to access them ... anyway that is a job for the future.)