I was just there this week off the Equinox, but we hired a private catamaran to sail over. I talked to someone at Soggy Dollar who came on the ship tour.
Evidently, they take you to the ferry dock, not to White Bay.
Then you take a bus over the hill to White Bay, where all the bars are.
So, unless someone else can confirm/ deny this, it sounds like yes, you can cheat and get to S$B without getting soggy. Not nearly as much fun, tho.