You are on the right path so far regarding the three easy jobs. For the fourth job, the portion you earned in Michigan is taxable to Michigan and only to Michigan. The portion you earned while in North Carolina is taxable to only North Carolina. You are exactly correct, if you were never physically in Michigan after establishing residency in North Carolina, the income is not sourced to Michigan even if that is where your employer is located.
To report this on the wage allocation in NC, you will want to split the W-2. Report the wages earned in MI or as a MI resident as "Not NC Source Income" and report the amount earned while residing in NC as "PY NC Source Income". The two should add up to the state wages in Box 16 of the W-2.
You do not qualify for a credit for taxes paid to Michigan. You only get this credit when two states are taxing you on the same income. If you allocate the income correctly from the job you had in Michigan and continued after your move, there will be no overlap of taxation. Michigan will tax you on their portion and North Carolina on theirs.
