So my advice is: practice your balance, and learn to deal with it. Or, pack some alternative footwear. A pair of Nike Air Hydrous water sandals ($60), for instance, could be useful when crossing streams or wet spots. They're made with a neoprene upper, and have a sole that's designed to grip on wet and slick surface. But we're not talking about "sticking" to wet surfaces, simply giving you a little better traction than a regular boot sole.
One other alternative is to pack along a slip-on set of spikes that runners sometimes use during the winter. Charlet Moser makes one such set, called the Spike Plus ($35). They're a soft, slip-on overboot that has six carbon steel spikes built into the sole. Very light - they fold up and go into a pocket - but they may give you the confidence you need to cross wet log bridges.
I hear that's a good hike. Have fun, and pack your raingear!