I agree with Ian. It probably won't happen, but it does happen to lots of people. That is the nature of hiking for a week. You might get a week of sunshine, or you might get rain 5 out of 7 days. I would bet on the sunshine, but be prepared for rain.
I've never figured out a perfect system for rain. I just carry plenty of extra socks and wear normal breathable trail runners. As Ian said, thinner synthetic tend to dry faster, but maybe thick wool socks are more comfortable. I tend to bring a variety for that reason and adjust accordingly. If it has been wet but we enter a dry stretch, then I dry out my shoes by putting in dry socks and essentially transferring the moisture to my socks (and then switching to another pair of dry socks). I've considered making socks out of the pack towel material for this exact purpose (as a means to dry the shoes). Thin liner socks would probably do just as well at absorbing moisture and drying out very quickly.
I could easily see a better system with shoe coverings. I've often thought about (and there has been some discussion of) bike shoe covers. You would have to adapt them a bit, but the idea is that rain would slide off your shoes, but you can quickly take them off (to let your feet breath) when it stops raining. That seems pretty slick, but most people just endure the rain.
Rain jacket and rain pants are a must. When it rains, it is usually cold, and if you don't keep your core dry, you can die.