I do believe it was Franklin who had the 13 pieces of the snake but I always thought the 13 pieces of the snake was printed with the slogan “Unite or Die” which had a slightly different context than the more commonly seen “don’t tread on me”.
I expect that both were kicked around at the same time during the revolutionary or pre-revolutionary war and Franklin made use of it as he saw best.
Of course I could just search for it, but what fun would that be…..I already researched it a long time ago, why go back and try to correct memory, right?