The “Texas dip” is a bow or curtsy that Texas debutantes perform, usually to great dramatic effect. It’s a difficult meaneuver that requires descending to the floor in style.
 
 
Answers.com 
Every debutante must perform a curtsy. This gesture is made as the young woman is formally presented. But debutantes from Texas perform the curtsy in a more dramatic fashion. By tradition, they do not merely bend the knees and bow the head but perform an extravagant maneuver known as the Texas Dip. Her arms are held out at her sides, she descends slowly into her curtsy, then leans forward over her extended leg, and lowers her head until her brow is close to the floor.
   
New York City Noticed
But debutantes from Texas have the most to worry about: By tradition they do not merely bend the knees and bow the
head but perform an extravagant Swan Lake maneuver known as the Texas Dip. 
 
Texas Presents: Presentation Gowns
And walking isn’t the sole challenge they face-the debutantes are expected to do a Texas Dip, which is a low-court bow where their head goes all the way to the floor. The girls practice the move for months with weights on their backs to simulate the weight of their train.
       
Houston Chronicle
Sept. 7, 2004, 5:39PM
The New York society scene may brag about the Chantal sisters, but Houston has the Cohn siblings. Morton and Bobby Sue Cohn’s daughters - Kirby McKool, Lindsay Holstead and Courtney Hopson - have been at the center of Houston’s society scene for much of their lives. As debutantes in 1990, they mastered the Texas dip, a curtsy that is unique to Lone Star society debs. Five years later, they chaired the ultra-successful Museum of Fine Arts Grand Gala Ball with their mother. They continue to be fixtures on the social circuit, drawing money and attention to their favorite causes.

Google Books 
Girls on the Verge:
Debutante Dips, Drive-bys, and Other Invitations
by Vendela Vida
New York: St. Martin’s Press
1999
Pg. 61:
No one know the exact origins of the Texas dip, bit it’s most likely a regionalized version of the St. James Bow—the bow debutantes did in England when they were presented to the Queen at the St. James’ Court. As its name implies, in America, the Texas dip is unique to Texas debutantes. More than anything else, tradition and a propensity for doing things in a grand manner probably inspire Texans to continue the custom while debs in the rest of the country do a simple curtsey. A Texas dip is quite literally a to-the-floor curtsey in which the debutante gets so far down on her high heels that her dress flares out around her like a marshmallow. As the coup de grace, she lays her left ear on her lap for a moment. The reason she turns her head is to prevent getting a lipstick mark on her virginal white dress.
 
3 January 1993, New York Times, “High Dudgeon In a Hampton,” pg. V8:
Texans Dip to Conquer
The Texas Dip is an over-the-top, forehead-to-the-floor bow perfected by debutantes from the Lone Star State. An amusing tonic for people suffering from debutantitis, it could be seen last Tuesday, for one night only, at the 39th annual International Debutante Ball at the Waldorf-Astoria. While other debutantes merely curtsied and received applause when formally introduced, all eight Texas debutantes elicited whoops and cheers with those dips.
 
You don’t need to be a debutante to do the Texas Dip. You do, however, need a white ball gown or a floor-length bathrobe to create the illusion of sinking into yourself on your way down. You also need a bouquet (any flowers will do), which you should hold with both hands below the waist.
 
Now, find the inner serenity that elicits your most confident smile, take a breath and raise the bouquet with both hands. When it’s at should level, let go of it with your left hand and slowly, as if you’re blossoming, open your arms.
 
Then you begin the descent. Cross your ankles without looking at them. Fluidly, effortlessly, bend the knees and sink down. (Think, “I’m melting” from “The Wizard of Oz.”)
 
Now comes the hard part.
 
While holding onto your escort’s gloved hand (you can practice with a desk or table edge), collapse onto your behind while smiling. Then, keeping your bouquet extended while looking forward, lean down with utter humility, bringing your face all the way into the skirt of your gown. Close your eyes as you go (think “dying swan”), and tuck in your lips so you don’t get any lipstick on your dress while you’re down there.
 
As you arise with the help of your escort, lift your head out of your dress, opening your eyes as if awakening after a hundred years. (Think “Sleeping Beauty.”)
 
Then, fluidly and effortlessly unseat yourself and unbend your knees, taking care not to move too quickly or too slowly. If you do, you’re likely to fall down.

7 January 1996, New York Times, “Where Texans Go For a Dip” by Bob Morris, pg. 33:
BECAUSE of the Texas Dip—a flamboyant nose-to-the-floor curtsy—the Lone Star State’s contingent at the International Debutante Ball always makes a big impression.