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I don't think it's a problem if you want it that way. I have a 4 syllable mn and I used to go by that. Some tried to shorten it and call me by a nn, but I'd correct them once and never have the problem again. Although you sometimes have to correct, most of the times it's ok. I mean if you introduce yourself as Elizabeth then most people will take it as such. There aren't that many that will shorten it.
But to tell you the truth I don't think there would be a problem with Alexandria. I know plenty Alexandras who use their full names, so Alexandria wouldn't be very different. If you insist at first that that's what you want the others to call her and correct them a few times I think it'll be fine.
And if she'll ever shorten it I don't think it would be because she gets annoyed by all the corrections.