The first step is to get a mailing address in SD that is recognized by the state as a legal address for driver's license and registration. Do this ASAP as your closing date is only a month away. We used Dakota Post
in Sioux Falls. There are many others, and they will have info on their websites on how to become a resident.
Dakota Post, as well as some of the others, will assist in registering your vehicles in SD. That is the second step.
No, you can't keep your Utah driver's license if you want to be a resident of SD so the third step is to go to SD and get new driver's licenses. That must be done in person because of the eye test and photo. Be sure to gather all the necessary documentation ahead of time. This is listed on the Dakota Post website. You will also need to fill out an affidavit stating you intend to make SD your permanent home.