That’s my set-up: low-cut, non-waterproof hiking shoes with wool or synthetic socks. Works well for me in almost all settings. Requires a tweak for frequent stream crossings. Two years ago, I accompanied Manfred & Sons on the first day of their Brooks Range Traverse. We crossed the same dang stream about 60 times on the way in. Then I turned around and crossed it another 60 times on the way out. 45F air temps, 36-38F water temps. In a few stretches when my feet were out of the water for 30 minutes, my feet would warm up. Still wet socks and shoes, but warm feet because we were hiking pretty fast.
But with stream crossings every few minutes? The flow of water through my shoes/socks would carry away so much heat, that my socks, shoes, the water in them and my skin would reset to 38F and warm very slowly from there.
My fix: bring plastic bags. Make a “dry suit” for your feet, albeit it a slightly leaky one. So it becomes a wet suit, but it still works. Some kinds of newspaper plastic bags work well. If yours do, buy a roll off your newspaper delivery person. Better, in general – large enough, tougher, somewhat flexible – are the plastic bags around loaves of bread – Wonder Bread, OroWheat, most breads in fact. Save them. Ask your friends to save them. Put them OVER your socks but inside your shoes before your first stream crossing. Even if there’s a small leak and your socks get wet, at least the water won’t FLOW through your socks so the wool/poly will maintain some insulation value like a wet suit does. And, for a mild conditions, your socks will stay pretty dry.
If you keep them for a while after the crossing, your socks will stay dry while the outer shoe slowly dries. If you remove them immediately, they will have keep your feet somewhat dry, but you’ll wet-out your socks and they will take longer to dry.
You can get more sophisticated about it and wrap the plastic bags over the top of your socks and secure them with a rubber band or a loop of small-diameter shock cord (I keep spools of it around for replacing tent shock cords and holding trash bags in place and stuff like this. PM me if you want some). Sized right, the loops is quite snug around your calf but looser when you slide it back to your ankle (or take it off) after the stream crossing.
Those same bread bags are good emergency gear for Fall or high-elevation trips. If you get an unexpected snowfall and you’re in your summer shoes, that plastic-bag trick is really effective in keeping your feet warm for the hike out.