There is a lot of public land in the Little Missouri National Grasslands of North Dakota but is more muley country than whitetail. There are definitely whitetails there but I'm sure it would take some scouting to find them. You can find big whitetails on other public lands in ND too but you really would need the knowledge from one of the locals who would be reluctant to give up their honeyhole. Including me, sorry. Haha. My 2 biggest whitetails(150-155") from ND have come from small pieces of public land.
Have you looked at the private lands open to hunting that SD has? There are some very large chunks of land open to hunting in the northwest part of the state. Not sure how good of hunting it is but I've looked at it on their website.