I was in a similar situation for a few years when I was with MSC and shipboard for 300 days a year. I switched my residency to a non-income tax state (NH), though not SD. I had to show an address briefly to get the license, but after that I just used a PO Box for everything.
You likely have to file a partial-year return in the state you left to cover the year-to-date tax obligation through the date when you moved to SD. And it shows that state that you left.
Other than that there’s not much to it. SD might not even have any personal income tax filing so you shouldn’t really need a CPA or a lawyer for state tax purposes if you’ve already established residency there.