You shouldn't be able to sell the speedo, since it's a quest item. The chest at the Pine Island beach where you jump into the water to meet the giant turquoise slime does have a speedo as well, and so does the one in Piratetown near the underwater entrance. Unless you already got a speedo in your inventory, then those chests give you 150 gold instead. Check your inventory again to see if the speedo is there anywhere, or maybe another one of your characters is wearing it? You need to equip the speedo to the Adventurer for it to work.