When I went into broadcasting, the thing I had to work on was putting the 'g' back on my 'endins'. Seriously, that was pretty much all I had to do, to bring my 'accent' up to Standard American Broadcast - which supposedly has 'no' accent.
But as others have pointed out - Texas is a BIG state. Big as France. Lots of 'regional' accents. There is definitely an 'east texas twang' for folks brought up in the 'piney woods'. But it's not that distinct in Houston. And Dallas has colloquialisms that are peculiar to a more 'Conservative' setting than Houston or say, Austin.
I think if you use a few of the common points we've brought up - 'fixin to', 'y'all' , 'put it up' , and make a point of dropping the 'g's occasionally in her speech - you'll get a feel for it. "I'm fixin to go to work." "Are y'all bringin the food with you?" "I was talkin to my friend, and he said he ain't goin." The 'g's simply aren't on the endings.
I completely agree with sounding more Texan when I speak with my family. I was living in upstate New York, rooming with guys from across the US. Answered the phone and talked with my father for half an hour. When I hung up, they were all staring at me funny. "What?" I asked. They burst out laughing, "Youse sounds like you waz inna movie or sumpin! Youze sounds so funny!"
My mom was a HUGE fan of the 'bless your heart' rule. You can say ANYTHING bad about someone, as long as you preface it with "Bless your/her heart" - "My sister, bless her heart, hasn't got a lick of sense, and can't keep her legs together." "My son, bless his heart, is a sandwich shy of a picnic."
I don't know if you can watch "The Big Bang Theory" - but the actor and character of Sheldon Cooper - is from Texas. The actor is a native Houstonian. He's constantly dropping his 'g's when he talks fast. That's a typical Houston accent, if you listen to him.
And yes... Dr. Pepper rules.