I cycle in flat walking shoes, Scarpa are my favorite.
If I'm on tour and the shoes I wore on the bike got soaked, I'll change into Crocs.
Crocs are very light, you could hang them from our rack thus saving pannier space, you can use them indoors or going to restaurants, etc.
I also own a couple of pairs of waterproof socks, nothing fancy, just the Aldi ones.
They are very effective, your shoes can be wet but your feet will be dry.