If you want TRUE period music (period to the 1870's) you would do well to listen to more of the Civil War era groups. Most of the songs that were heard during this time period would have been around just prior to, during and after the war. What was popular during the 60’s was, largely, still popular during the 70’s. Minstrel songs, parlor songs and even some classical pieces would all have been heard in what we consider the “old west.”
“Cowboy songs” as we think of them, are largely a 20th century invention, with most of the ones we think of as “period,” being written between 1890 and 1920. The songs that were actually “sung on the trail” would have been earlier tunes.
True period music would be played on original or period reproduction instruments without steel strings. Gut strings were the rule during the 1870’s. It’s extremely rare to hear any “cowboy” recording done in this fashion. Most artists prefer the more modern steel strings.
There are several good groups from the Civil War genre that do perform period songs on period instruments (with the correct strings). Groups like The Camptown Shakers and the 2nd South Carolina String Band are a good place to start. There are several others, but if you’re looking for TRUE period correct music for the 1870’s, you’ll need to by-pass the Cowboy genre and go a bit earlier.