Once you begin living in Michigan, you will need to pay MI state income tax. You pay income tax to the state that you LIVE in even if you are working remotely for a TX-based employer. Find out if the employer will withhold MI tax from your paychecks. If they will not do that, then you should make quarterly estimated payments to the state to avoid a big state tax due at tax time.
TX has no state income tax, so you do not have anything to file for TX.
https://ttlc.intuit.com/questions/1901227-how-do-i-file-a-part-year-state-return
https://ttlc.intuit.com/questions/1901560-how-do-i-file-if-i-moved-to-a-different-state-last-year
https://ttlc.intuit.com/questions/2895983-how-do-i-allocate-split-income-for-a-part-year-state-retur...
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