Yes, I am serious,
you do sound fussy, precious and uptight correcting everyone!
Just be thankful people are talking with you 
Correcting them all the time is very off putting, especially when abbreviating someone's name means you're "in" the group.
Why do you insist on correcting people when it's a friendly gesture? Don't you want to be included, to be one of the team? If not, why not? And where would you rather be with your 'full legal name', as you say yourself.... in front of a judge?
Just go with the flow, see an abbreviation or a nick name as a compliment.
The only Alexandra I know about is Princess Alexandra, who was about 100 years old and fusty with it. It's not a name I'd chose over Xandy or Alex if I was working closely with a group, and they liked me enough to use a NN. I certainly wouldn't correct them every time.
Maybe people you work with feel like me, they think Alexandra sounds snobby and overly formal, (because of the royal connection) and prefer to think they work with someone who actually is a human being who isn't afraid of getting stuck in, and doing some work.
I've been called all kinds of NN, some of them nothing at all to do with my actual name, and I think of them as a rite of passage/ badges of honour.
I am very happy to muck in when working in teams without ceremony or formality though. I work mostly in the stem field with blokes, and there's no question of anyone being up themselves or prissy.
A name's just name - it's not you.
Maybe you've never thought about how others view you, or what your name means to them? If they're always using a NN for you, maybe it means they don't like using your legal name I guess, and want you to mesh into the team without standing on ceremony.
Four syllables is a lot, ya know, especially if everyone else is know by a NN.