New Jersey applies a version of the convenience of the employer rule, but it does not apply to South Dakota.
https://www.state.nj.us/treasury/taxation/conveniencerule.shtml
If your domicile is in New Jersey, and you have been temporarily away, your income is taxed in New Jersey (NJ-source). To change domiciles, you must not only set up a new domicile but take active steps to abandon your prior domicile. Until you abandon your NJ domicile, you are an NJ resident and all your world-wide income is taxed in NJ.
After you establish a new domicile, your income will only be taxed as NJ-source if your new state also uses the convenience of the employer test. Since South Dakota does not use this test, your income after you change domiciles will not be taxed in NJ. When you prepare your tax returns, you would file a part-year resident NJ return, and you will need to manually allocate your income (Turbotax can't guess for you).
After you relocate, income will only be considered NJ-source if it is earned or paid while you are physically living or working in New Jersey, such as if you are required to be on site for training or meetings periodically.
If you are not currently domiciled in New Jersey, your income for the whole year would not be NJ income, unless you are in one of the other states that also uses a convenience of the employer test.
