Yes, you will need to file part-year resident returns for both states.
You'll file the Indiana return and only report the income that you earned while a resident of Indiana. Since Michigan and Indiana have tax reciprocity, you do not owe tax to Indiana for the time that you lived in Michigan - even though you worked in Indiana.
You'll also file the Michigan return and only report the income that you earned while a resident of Michigan. This income is only subject to tax in Michigan, so you won't qualify for a credit for taxes paid to another state. You should be refunded by Indiana for the tax you paid in Indiana on the income you earned while a Michigan resident.
Since the returns are not dependent on each other, you can file them in any order. I recommend requesting that your employer stop withholding state tax for Indiana. If they'll withhold for Michigan, that would be best. If not, you should make estimated payments to Michigan to avoid a balance due at tax time.
