Your residency is generally where you maintain a permanent place of abode with the intent to remain there (or return there after an absence, such as school attendance, vacation, temporary employment,extended vacation). Where you are licensed to drive is only one factor, and does not define your residency status.
If your wife lives and works in SD year-round (other than short absences) and you attend school there for 9 months of the year, you are most likely SD residents.
However, you should consult with SD tax authorities to make sure how that state defines "residency". Registering to vote, full-time employment, "permanent" place of abode, owning property, having a state driver's licenses, are all factors that come into play.
https://ttlc.intuit.com/replies/3301816
