My approach (each to their own):

Fast-drying (non-waterproof) trail runners or similar.

If crossings are frequent or walking in streams, just leave shoe inserts in and a thin liner sock on and go for it. If this is likely to get cold, some sort of neoprene sock is very nice, sirius storm socks or neosocks or the like.

If crossings are infrequent then take out inserts and take off socks prior to crossing, just bare feet in shoes. Once on the other side, dry feet, squeeze out what water you can from shoes. Put inserts back in shoes, put on dry socks, and cover with gore-tex socks. This way your feet stay dry while you walk the shoes dry.

Goretex socks and/or neoprene socks are somewhat dual use as they're good for snow as well as stream crossings.