There's no discussion of what is considered "dignified" but I do know that when the code was written in the 1920's it originally stated that the flag should be destroyed "in private." Apparently there were enough patriotic organizations who thought otherwise and wanted to make a ceremony of destroying flags that have served their purpose. I am in favor of using this opportunity to teach people about the history, symbolism, and significance of the American flag, and why it should be treated with respect. But as I said in my earlier note, dismembering the flag by cutting the stars from the stripes is, in my humble opinion, completely inappropriate.
To bring this back to a Civil War era connection, there would have been no such process for destroying old flags. On board ships I would not be surprised if the US Navy followed the British custom of using the material from old flags as cleaing rags. Procedures covering respect for the Stars and Stripes were a late 19th and early 20th century innovation.
Nick