The problem with avoiding the question is, it tends to make a bigger deal out of it. If you tell me you're from Lavenderland, I'll probably think "Oh, Lavenderland, I hear they have X there," and I won't think about it any further, unless maybe one day I really need to know the Lavenderese word for pickle or something
In most cases, I wouldn't see you as just "the girl from Lavenderland," any more than I'd see you as that librarian, or the girl in Florida, or anything like that. Anyone who does find out where you were born and makes a big deal out of it, and can't see beyond it probably isn't worth knowing anyway.
If you make a secret out of it, people will want to find out why it's such a big deal. Since "where are you from?" is generally such a simple question, to give an out of the ordinary answer really really draws attention. People will want to know where you're from, why you're hiding it, what happened that was so terrible that made you hide it, why you left, etc. If I asked "where are you from?" and got "I'll let you know when we get to know each other better," I wouldn't be able to stop myself from focusing on it! If you don't want a big deal made of it, you're much better off just treating it like any other fact, and not making a big deal out of it at all.
I do think "Yes, but I'm much happier here" is perfectly fine. If someone asks where you're from, you can use something similar. "I was born in Lavenderland, but I'm so much happier here. I really love the *insert distracting bean dip about Florida and your interests here*"