Do we file married but separate? Does my husband file in both states then or does he just file in Minnesota?

For federal you probably will do best to file as Married Filing Jointly (MFJ), but if in doubt try running the numbers as Married Filing Separately (MFS) and see which works best.

For WI if your husband has no Wisconsin income and didn't live in WI at any time during the year then you can file as MFS, since they do not require that you use the same filing status as on your federal return, and your husband doesn't need to file a turn with WI at all.

For MN they do require that you file using he same status as on your federal, so you would need to file as MFJ and report both your incomes and figure a tentative tax based on all income. But you then complete form M1NR, on which you show that your income is not MN based, and take a credit that eliminates the portion of tax attributable to your income. Bottom line is you don't pay MN tax on your WI income, but the tax rate for your husband's tax is based on your combined income.