On cotton, you can look up records of cotton gins in the area you are depicting. If you can find cotton gins, there would have been cotton clothing available for pretty much all classes.
As far as hunting shirts go, I suspect that at some point during this time period the capes became optional and possibly even fell out of style eventually. For instance, this portrait of Stephen F. Austin reportedly from his 1833 trip to Mexico City depicts him in a capeless hunting shirt and matching trousers or leggings.
If you look at the Davy Crocket Almanacs (published in Nashville beginning around 1835) there are more depictions of him wearing a hunting shirt without a cape than there are of him wearing one with a cape. There are also plenty of depictions of normal coats, etc. being worn, so that isn't to say that a hunting shirt is mandatory.
This appears to be one trimmed with fur.
From the 1839 one. Based on the fact that it is belted closed, I would call it a hunting shirt. Could be the artist didn't want to draw fringe on it, or it could be just a regular coat and he happens to be wearing a belt for other reasons.
This last one is from 1847, so well after the time period you are interested in. There appears to be fringe on the bottom, but lapels like a coat.
I would expect these Almanacs to depict frontier styles at the time, but some of the later ones may have been published back east, so I would rely less on their illustrations. However, if they are accurate it would lend support to my theory that hunting shirts got more and more coat like through the early to mid nineteenth century, losing the cape and gaining a tailored cut somewhere along the way, until they essentially became the western fringed jacket.