"That's All Right," written by Crudup, was released on Victor in 1946, hence is not public domain.
Some of the lyrics were taken from Blind Lemon Jefferson, 1926, on a Victor label, also not in public domain.

At BMI, the work is Crudup work no. 1479466
Unichappell and Crudup were the two publishers.

You would have to get permission through BMI.