adjective-born or noun-born?
Solution 1:
The choice between what the poster describes as the adjective and noun forms for X in the phrase "X-born" may reflect the originating notion: "born a French national"/"born a native Texan" on the one hand, or "born in France"/"born in Texas" on the other. I've encountered both forms in different settings, but (at least in published writing) usage has gravitated strongly toward conflicting results: "French-born" and "Texas-born."
The earliest instance of any of these phrases that a Google Books search finds is from Randle Cotgrave, A French and English Dictionary, Composed by Mr Randle Cotgrave, with Another in English and French (1673):
Estrayeres : f. Escheats; the goods of strangers dead without French-born issue; and of bastards, dead intestate, or without issue.
Soon thereafter, from a letter of January 28, 1680, reproduced in David Jones, The Secret History of White-hall, from the Restoration of Charles II down to the Abdication of the Late K. James (1717):
Lastly, The Duke [of York] was to take care, that no Popish Clergy or Laity should be employ'd by him, but such as were in the French Interest, and trust his main Secrets with none but such as were French-born Jesuits ; on which Conditions he was to have a considerable Annuity of six hundred thousand Crowns, and extraordinary Sums when necessary, and the Circumstances of things did require to carry on any of the foremention'd Points, even to what he pleased himself to demand.
Though the number of matches isn't large, it does extend over the centuries, and includes an interesting example where the adjective form applies to both country and city. From Mrs. Gore, "The Red Man" in The Courtier of the Days of Charles II. With Other Tales (1839):
“No such thing!” interrupted Balthazar. “The ironwork does honour to a trusty workman, who must have served his time to a master-mechanic of the cité. The hand is that of a woman, French-born,—Parisian-bred.—The victim was, in short, one who lived and died almost within sight and sound of the very spot where we are standing!"
Various Google Books searches for the years 1600–2008 turned up just one example of "France-born." In part this may reflect the trouble such searches have with hyphenated phrases, but that same disadvantage didn't prevent Google Books from finding a number of instances of "French-born." From Sarah Thomason & Terrence Kaufman, Language Contact, Creolization, and Genetic Linguistics (1988):
Though in England the settlement appears to have been more intensive than elsewhere Norse probably lasted no more than two generations after 955. In Normandy, for example, the first France-born generation [of Norse settlers] had almost only French names.
Note that this instance involves something of a special case: Although the Norse settlers in question were indeed "France-born" (that is, born in the geographical region of France), they were not in any practical sense "born French."
Google Books search results for "Texan-born" and "Texas-born" show a very different tendency. Matches screened for relevant instances of these terms come our heavily in the latter's favor—57 matches for "Texan-born" between 1900 and 2008, and 221 matches for "Texas-born" over the same period. Interestingly, a significant number of the "Texas-born" results come from Texas Monthly magazine and from Alcalde, the University of Texas alumni magazine. This suggests to me that Texans themselves favor "Texas-born"—as indeed my Texas-born parents and siblings do.
On the other hand, the numbers since the turn of the century are closer—30 for "Texan-born" and 53 for "Texas-born"—indicating that a significant number of people today may be taking their cue from "French-born" and similar phrases.