I always bring a pair of lightweight shoes or closed toe sandals when backpacking. I find that having a different pair of shoes to put on once I get to camp makes a world of difference to my feet. It lets my feet relax and gives both my feet and my boots more time to dry the sweat out. I also use these for crossing wide streams.
If you don't want to pack extra shoes, the next best thing is to take off your boots and just wear your wool hiking socks. The socks will give some protection to your feet and they actually grip slippery rocks better than the rubber soles of shoes or sandals. If the stream bottom is sandy or muddy just cross in bare feet - socks won't help in those cases.
On the other side of the stream, wring out the socks and hang them on the outside of your pack. Dry your feet off well and let them air dry until they no longer feel clammy. This is a good time to apply any necessary treatment to your feet (friction blocker, mole skin, athletic tape, talc, etc) as they will be clean and dry. Put on dry socks and your hiking shoes and continue down the trail.
If you encounter another stream just put on the socks that are already wet for crossing the stream and repeat. Be sure to dry those socks out well on the outside of your pack and at camp so that you can still use them for hiking if needed. They'll get caked with dirt when you walk up the bank of the other side of the stream so you'll probably want to wash them out well in the last stream you cross so that they're cleanish for later.
As for my pack, I always have everything in my pack in either ziplock bags or trash compactor bags. Be sure to put your hiking shoes in a bag before crossing the stream and put them in your pack so that you have two hands and so they don't get wet should you fall.