There are two possible sources of the wetness. One is condensation on the outside of the chocolate, caused by keeping the dipped berries in the fridge, then exposing them to room air while they’re still cold. The tiny beads of water leach sugar out of the chocolate and form a syrup on the surface that never quite dries. The chocolate is then permanently dappled with shiny, sticky syrup.
The key to avoiding condensation is to not allow warm room air to come in contact with the cold chocolate. One solution is not to refrigerate them at all. If the chocolate is tempered, there will be no need for cold to get it to set. And if they’re to be eaten within a couple of hours, the berries should be fine at room temp. But if you do need to refrigerate them, put them in an open food storage container, stick them in the fridge to get cold, then quickly reach in and snap the cover on the container. Now here’s the important part: do not open that container while the berries are still cold. Leave the container out at room temp for at least an hour before opening it. When you finally do expose them to the warm air, the surfaces shouldn’t be cold enough to provoke condensation.
The second source of moisture is more problematic. The sugar in the chocolate draws juice out of the berries themselves. This collects between the berry and the coating, and will eventually leak out as a pink syrup. There really isn’t anything you can do about this other than eat the berries before the syrup gets to the leaking stage.