I've walked hundreds of kms in sandals and socks in the rain by now. Never had a problem because of that.
Merino wool socks keep the feet warm even when wet. Even if you put cold, soaking wet merino socks on your feet, they only feel uncomfortable for a few seconds. As soon as you start walking, feet get warm again, and if it doesn't continue raining, they dry quickly on your feet, also, if the shoes allow for it (more difficult with heavy boots).
Never had a blister because of wet feet/socks.
If you want your shoes to dry more quickly, you can put old newspapers into them over night. Ask the hospitaleros, they usually have some. It's also good to take out the insoles for drying.
I think wet feet are inevitable on a hike with lots of rain. Even gore tex boots and waxed leather boots will soak through after a while, and will need much more time for drying than something more breathable and open (like a hiking shoe with mesh parts or a sandal). No shoe/boot is 100% waterproof, and if they are, your feet will get wet and hot from sweat, which is more problematic than rain or dew.
I just accept that I have wet feet sometimes when hiking. On some tours I walked through constant cold rain for days, with soaking wet socks. No problem. But of course that was in sandals, not shoes.
I always change into dry socks when resting in a dry area. For example in a bar, I change into the dry socks in the bathroom. That way the feet can dry off a bit. Then when I continue my walk, I put on the wet socks again.
In cold/wet weather, I also carry 1-2 pairs of socks more than in summer. One pair is always kept dry, for the evening.