First rule is to stay comfortable, warm & dry. Gear up!
Mud boots -- similar to rain boots, but better made and generally have a grippier sole. These allow you to walk into 6-10" of water while your feet stay dry. Wear rain pants over them and you won't have to worry about damp feet.
Buy them large enough to wear two pair of warm socks. Because there are no laces, the boots are loose on the top of your feet, and the second pair of socks not only keep you warmer, but will keep your feet from slipping around inside the boots. (I also added a pair of Dr. Scholl's gel insoles).
I also wear a latex glove (actually nitrile) under a thin glove on my off hand so that hand doesn't get wet when I pick up a wet disc. You can pick up a pack of six at the dollar store.