Some like to muddle a bitters-soaked brown sugar cube until "smooth paste" to make this cocktail. Depending on your sugar cube, even with a splash of soda...
Tried with Appleton Estate 12 year and Rittenhouse Rye. It's okay, but I prefer a regular sazerac. I'm starting to think I only like Jamaican rum in mai tais.
I love a Sazerac.
Your issue may have been the Appleton. We are big rum drinkers and find that Appleton is a bit tame for a Jamaican rum. I would go with a Hamilton Black or Myer's Single Barrel. The Jamaican funk is needed to balance the rye.