This figure is true of all characters in the film, from those of relatively high station, down to outlaws and stable boys.
For added color, the Coen Brothers, who wrote the film, are known for being meticulous in their research regarding period dialogue in their scripts, so I would expect historical accuracy here, however, I obviously want to ask actual historians about this.
Part of what makes True Grit so enchanting as a novel (1968) is the character of Mattie Ross. Readers of the novel (and to a lesser extent, viewers of the 2010 film) get to know two versions of her: (1) Mattie Ross the young adult who pursues her father's killer, and (2) Mattie Ross the aging woman who tells the story.
When you read the book, it's a bit like listening to Great Aunt Mattie tell stories about the Wild West day of her youth. Sure, there's Cowboys and Indians, but it's crazy old Mattie who's telling the story and anything anybody says sounds a lot like old Mattie. You're left wanting to know more about Mattie but also wondering how much of her story could really be true.
When you translate this novel into a movie, the stakes are different. You're no longer hearing Mattie tell you what Rooster Cogburn said. You're watching someone called Rooster who acts like a rough-and-tumble lawman but talks like an old woman. The effect can be jarring, to say the least.
That said, anyone living in Arkansas in 1873 would have been known and probably used contractions in everyday speech. To take a familiar example, Mark Twain (i.e. Samuel Clemens) published the Adventures of Huckleberry Finn in 1885, and it's set along the Mississippi River in the 1840s. Twain included an explanatory note saying that he was trying to replicate different kinds of speech the he himself was familiar with, and since he grew up in Missouri, we've got good reason to trust his expertise. Here is a brief snippet Twain gives us of dialogue between two people from Arkansas—Tom Sawyer's Aunt Sally and Uncle Silas Phelps (with Huckleberry Finn impersonating Tom Sawyer).
So people would have generally used contractions. In the 1880s, Mark Twain used contractions to make his Arkansas characters believable. In the 1960s, Charles Portis did in fact use contractions, but he adopted an awkward form of speech to help separate the story from its narrator, making both more interesting as a result. In 2010, the Coen Brothers apparently avoided contractions (and I'm here trusting OPs observation without checking myself). My impression is that the Coen Brothers opted to follow the speech of the 1960s novel more closely than the historical speech of the 1870s. I suspect you'd have to ask the them what their priorities were to figure out why.