Did you remove the first two characters of each sequence before calculating the Shannon entropy? I don’t know how much the predictability of the first two characters would reduce the total entropy score.
I am not familiar with Gold/Kasami coding, and I wouldn’t try it unless I could find reference to anyone using those codings for sequence to sequence character or language stuff.
I can’t really make any decent recommendations for statistical analysis beyond very basic stuff, such as a scatter plot of character position against ordinal number. Then maybe to count the different bigrams and plot their frequencies.