Thank you for all your guys' input. I use OnX religiously and it is a great rule. But like you said there are discrepancies between what is out in the field and what the maps say. I always lean on the side of caution and don't push my luck if something looks private but is shown public. So with this 33' section line, that is the edge of a private piece and a public piece that has a two track road on it? I guess I am confused with that.
I hope I can help without resorting to surveyor/legalese jargon. Two separate issues are in play here: legal access and physical access.
In ND and SD the public can (with a few rare exceptions) travel within 33' either side of a section line, which is a land boundary and a legal matter. Ownership is irrelevant. Therefore, you can travel along the edge of private property as long as you're within that 66' corridor along a section line, and corner crossings are a non-issue. I've traveled and surveyed hundreds of miles along section lines in ND and never had a landowner problem. They know the law and seem to leave folks alone. The trick is knowing whether a fence is on or near a section line. Most are close, and owners I've encountered think the fences are close enough. Sometimes you'll find double fences with up to 50-60 feet between them, in which case the section line probably runs more or less up the middle.
Regarding physical access: Some section lines have trails along them, some just have fences, some have a thin uncultivated strip, and a lot have fully-built county roads and highways. Just be smart. Don't drive where a farm/ranch trail doesn't exist, stay out of cultivated fields, don't make ruts, etc. Be a good steward of the land. If I have to walk a mile I figure it's a bonus--I can always use more exercise.
If you get hassled, keep in mind that you are innocent until proven guilty. An owner that wants to press trespass charges will have to prove your guilt, which will probably mean hiring an expensive surveyor--not worth the money. OnX and other Geographic Information Systems do not show exact legal boundaries, and they are not based on legal land survey data. They are merely graphical, and they have no legal standing whatsoever as far as geographic location. In general, fences, roads, and trails tend to be within that 66' corridor, and such on-the-ground evidence holds more weight in a court than OnX.