For tax year 2024, you'll file a part-year resident return in each of the two states. Your income prior to the move will be taxable by both MN and WI; your income after the move will be taxable only by MN. On your part-year WI return, you'll be able to claim a credit for the taxes paid to MN on that portion of your income.
How do I file a part-year state return?
On your W4, you should have taxes withheld only for MN.