I'm a fan of winter cycling shoes. Shimano has some neoprene cycling shoes (I use SPD).
I've used shoes similar to the Shimano MW5, and like them for cold or wet. Not perfectly dry, but not bad.
Also get a boot dryer (or 2 or 3) for any place you wish to dry your shoes, socks, gloves, etc, over the course of several hours, or overnight.
Most have a small heater coil at the bottom and rely on convection, although some apparently have an actual fan too.
If you have ducted heating with floor registers, you might try simply sitting the boot dryer on the vent without plugging it in.