Don't feel bad about not being a native Texan. Here in Austin, I frequently find natives less Texan than I am. Besides, Texas' greatest heroes were not native Texans.
I have lived in Dallas, Houston and San Antonio. Austin is the first place I have found limp wristed, bed wetting "Texans" with Texas accents. I still cannot get used to it.
Oh, I don't. The thing is, I've had the "Texas" mindset for most of my life, but didn't know there was "a whole 'nother country" full of like-minded folks.
After moving here, my dad mailed me a certificate he'd gotten from Texas First Families that certifies that my family qualifies for that fine distinction. Turns out that my GG grandfather was born here prior to statehood.
I've cautiously mentioned it to native Texans, who've all given me a sort of Texas style thumbs up about it.
As they say, "I wasn't born in Texas, but I got here as fast as I could."