I think there are two schools of thought on this situation: (1)You can take heroic measures to keep your feet dry although you'll probably fail or (2)You can go with the flow, literally in this case, and just let your feet get wet.

Unless it's really cold and/or I'm travelling in snow I prefer the second option. When I did the JMT in an extra wet year ('96) I had to cross many streams every day for the first 180 miles or so. My feet were constantly wet. No sooner did they begin to dry out than it was time to cross a stream again. Taking them off was too time consuming for me. I used synthetic shoes with thin nylon socks. I controlled blisters with lots of waterproof tape. Although wet, my feet were seldom cold and I was reasonably comfortable.

Next time I may try thin neoprene socks. They're designed to get wet.