Hmm, I'm not sure it would work to be honest. I think the reason for choosing a nickname is that the given name is too long or too formal sounding for everyday use. However
Alexandra is still a long name in itself. Also it's so similar to
Alexandria that I think it would create confusion for other people about which was the "correct" name.
I personally think if you love the name
Alexandria (and have done for a long time) then use that but go for something more obviously nicknamey and less formal as a nickname (if you need to) e.g.
Lex,
Lexie,
Alex,
Alexa (I like this as a full name but it can work as a nickname),
Ria,
Sandie etc.