Two choices:
a) SPD sandals, no socks.
Both skin and sandal dry out quite quickly when it stops raining. Since the circulation in the feet is pretty much unrestricted, it's not as cold as you might expect, besides which it encourages you not to be nesh.
b) Winter boots, that come up the ankle a short way, plus ankle seals to redirect water running down the leg to the outside.
Drysuit ankle seals, with the narrow end against the skin of the leg and the wide end over the outside of the boot, are best, but you can do the same with the wrists cut off a pair of washing up gloves. If you wear longs, they should go over the outside of the top of the seal.
Waterproof socks plus ankle seals will keep the foot dry, but you'll still have soggy shoes the next day.
Without the seals, water will run down the leg and in the top, the same as shoes.