I don't know about SC humidity specifically, but without electricity, the only option I know of is dessicant dehumidification, such as Dri Z Air or something like it:
DRI-Z-AIRÂ® DEHUMIDIFIER - MOISTURE ABSORBER - MADE IN THE USA
I haven't used the Dri Z Air myself, but I know that it removes water from the air, the question is how much. The electric dehumidifier in our boat accumulates a substantial amount of water, so the capacity of this system looks suspiciously low to me especially if you plan to leave the trailer unattended for a long time. Putting several of the units in the trailer would presumably help. For perspective, prior to putting an electric dehumidifier on our boat, locations with limited air circulation would develop mildew spots. After using the dehumidifier, no such problems. Another boat owner where we moor returned to his boat after several months away to find the majority of surfaces inside the boat with substantial amounts of mildew on them. This result may have been exacerbated by rain leaking into the trailer, but I do not have information on that. Suffice it to say, that in very high humidity environments, this issue can get seriously annoying. Perhaps someone else here has specific experience with dessicant dehumidification, but if I were you, I would be careful.
In any case, storing your cushions elsewhere is a good idea. Checking on the trailer periodically until you learn how things behave, and leaving all drawers and doors including the fridge ajar would be another one. If you notice mildew starting, it will be time for a plan B.