"I'm Going to Leave Old Texas Now" is a variant of "The Trail to Mexico." See variants in Fife and Fife, 1969, "Cowboy and Western Songs," esp. text E, p. 182.
In turn, the melody is inherited from "Early, Early in the Spring" (and other titles), Campbell and Sharp, 1917, "English Folk Songs from the Southern Appalachians," discuss English precursors (not seen, reference from Randolph, "Ozark Folksongs," vol. 1, p. 333). Belden refers to a 17th c. broadside, "The Seaman's Complaint," "printed for Charles Baker" (British Museum C. 22, f. 14, p. 175).

The first verse is:
It was early, early in one spring,
I was pressed on board to serve my king.
I left my dearest dear behind,
Who oft times told me her heart was mine.

The song of disappointed love and its cowboy derivative, in many variants, is widespread, from Newfoundland to West Virginia to the Ozarks, Mississippi and Texas.
Compare "The Sailor Boy's Bride," in the Bodleian Library.

The mention in a Steinbeck novel of Indian or Spanish precursors is fiction.