You face several hurdles, some of which to overcome you may face serous penalties.
It is possible to put a PO BOX or a mail forwarding service as your mailing address when getting a drivers license. They will also send a renewal out of state, for example to a kid in college. But they will want a "real" address for your in-state residence. This is even more true when dealing with a real-ID compliant ID card.
This is how South Dakota does it for full-time travelers
To be eligible for a South Dakota Driver License, you must have a
physical residential address located in South Dakota, live full time
in an RV/camper, or travel full time for work (for example, traveling
nurse, truck driver, etc.).
If you use a mail forwarding service, you will be required to complete
a residency affidavit, including a perjury statement indicating the
answers you provided on the form are true and correct.
This is the residency affidavit
there is one minor note:
PLEASE NOTE: South Dakota Driver Licensing records are used as a
supplemental list for jury duty selection. Obtaining a South Dakota
driver license or non-driver ID card will result in you being required
to report for jury duty in South Dakota.
and then the words you will be swearing to:
“I declare and affirm under the penalties of perjury (2 years
imprisonment and $4000 fine) that this claim (petition, application,
information) has been examined by me and, to the best of my knowledge
and belief is in all things true and correct. Any false statement or
concealment of any material facts subjects any license or ID issued to
and by the way:
This form must be signed in the presence of a notary public (virtual
notary will not be accepted) or a South Dakota driver license
So on top of the potential tax fraud issues with your old state, and any other issues with your wrong state, and potential state and federal problems with voting in the wrong state, South Dakota could send you to jail for two years.