I found that on the midwest section of the Northern Tier route (Minnesota and North Dakota, for me, in particular), the specific NT route on the maps wasn't anything special. Other county roads or state highways worked just as fine or better in places. I veered off the maps and took my own route from Minot ND to Fargo, and again from Fargo to Minneapolis, and it worked out really well. You'll have to feel out your comfort level as you go, but with a smart phone with maps, paper maps from a gas station, and advice from people in towns, it should be no problem. Water, food and camping are usually available in small towns only anyway, so as long as there is a small town, it will probably have those things. Camping in small town parks seems to be free and OK across the midwest (I've had great experience with it), so I would recommend going off the NT if you want.
From Iowa or Minnesota you could go straight west on into South Dakota (lots of small towns along the way). Getting back up to the NT from Mt. Rushmore might be tricky as the towns are more sparse, but you could take Hwy 85 north to catch the NT near Dickinson. It might be worth it to go through Glacier NP, Sandpoint ID and the Cascades Mountains in Washington state. Sounds like a beautiful trip you're planning!