You'll find birds on public in SD if you have a good dog. after the first month of the season, hunter numbers start to diminish. Couple that with colder weather and crop harvest completion, and you'll see birds grouped up a little better, especially near heavy cover. I would target the areas between US hwy 281 and the missouri river to narrow your focus. I don't get out that way much, but the times I've had, we've shot birds every time on public lands. a good dog that doesn't mind pounding cattails is always a good idea.
Bird numbers seem to be up throughout the state; we hunted a 160 acre WPA on saturday for about 1.5 hours in the south eastern part of the state, with a very inexperienced dog, and we flushed 17 birds. with the wind and dry conditions I consider that a win.