Texas is a thing unto itself, that's a natural fact, because of the peculiar way it came to be; but it was not created
ex nihilo, that is, called forth out of nothing. There's a certain insistence on singularity in most Texans' characterizations of the place: "it's
not a Western state and it's
not a Southern state," they say, because to say "it is both part of the South and part of the West" robs it of a
soupçon of singularity.
Historically, economically, politically, and culturally, Texas has been part of the South since it joined the Union in 1845 (even that is peculiar, since it is the only sovereign nation to become a state by treaty, that treaty guaranteeing it certain unique privileges among states). It was one of the eleven states of the Confederacy. African Americans are pretty clear that Texas is a Southern state: they celebrate Juneteenth Day, the day that the Emancipation Proclamation finally became public knowledge in Texas.
The rest of the world has, until recently, treated Texas as a Southern state. And culturally, Texas has always had its eyes fixed to the east. Its manners, mores, and social institutions belong to the South.
The War of Northern Aggression
This would be especially true of East Texas, where I grew up. The hours I spent in the reading room of the public library in Houston, were spent underneath vast murals depicting the life of Confederate General Robert E. Lee. I never heard the words "Civil War" to describe what is known in the South as "the War Between the States." When in school we studied that war (sometimes referred to as "the War of Northern Aggression"), we were reminded that what we knew as the Confederate victories of First and Second Manassas, were known to rest of the world as the ignominious Union defeats of First and Second Bull Run. My high school's mascot was (and as far as I know, still is) the Johnny Reb holding aloft the Stars and Bars, the school colors are gray and red and black, and until very recently its admissions had been limited to members of the "white race".
This is not to say that the South is more racist than the rest of the United States (it is not: one of the surprises of going outside the South is the discovery that the rest of the United States are profoundly racist), only that its racism falls under the historical shadow of Jim Crow. I think Texas passes the "if it walks like a duck..." test as a Southern state with, ah, flying colors.
Y'all
Regarding accents. Texas is bigger than France. It has five regions which are distinct geographically, economically, culturally, and linguistically. So distinct in fact, that the Treaty between Texas and the United States guarantees that Texans may, at any time, hold a plebiscite for the purpose of splitting into five states.
There is no such thing as a "Texas accent". There are five regional dialects. Not surprisingly the East Texas dialect resembles most strongly that of its neighbor to the East, Louisiana. The accent most people associate with Texas would be that spoken in West Texas.
My own accent has only traces of my native region. Mostly I lost the diphthongs. Curiously the expressions are intact, and with my new Mid-Atlantic accent, they can sound comical, e. g., "I'm fixin' to do it," or "Let's get some Liquor!"
Faulkner drives some Southerners crazy. He (and Tennessee Williams) made up a South which, however true it may be in spirit, is a persiflage, an exaggeration, and something of a burlesque. To mistake the South, even the Old South, for Yoknapatawpa County, does some violence to the quotidian facts. Mostly Faulkner reinforced the worst stereotypes about Southerners, and made life that much more difficult for them. Faulkner told fine tales, but they're tall tales. I believe that they are very accurate in capturing the spirit. And finally, yes, in some respects we do live up to the stereotypes: I did grow up in a big house and everyone, including adults, referred to my father as "Daddy," (but not "Big Daddy"!).