You need to visit the Texas Civil War Museum in Fort Worth, mainly because it is a bang-up collection of artifacts from many States, not just Texas, but because they do have the flag of the 5th Arkansas that was carried at Murfreesboro. It is a full-moon flag with a quite glorious history.
I tend to believe that the flag of the 5th/13th Arkansas captured at Franklin has survived but as yet I've not run across it.
P.S. It was not on display when I visited but you can make an appointment to see it.