In this case you are correct. You will file part-year returns for both Missouri and Mississippi. You pay tax to MS for the income you earned while living there. It is not MO nonresident income because MO does not tax the telecommuter (there are some states that do, but MO is not one of them).
On the MO part-year return, you will need to allocate your income to tell MO how much of the income was earned prior to your move. You will claim that income in MS, and you will pay tax on the income in MS.
Your employer is not correct. They should withhold your tax to MS and not MO. How much income tax you pay to MO vs. MS is irrelevant. You don't pay tax to MO, period. By not withholding MS, you will face penalties in MS next year if you do not pay enough in. It may be inconvenient for the employer to set up an account with MS to send the state tax withholdings, but they are required to do so.
St. Louis is a situation that cannot be arranged on the Missouri return. You will need to contact St. Louis revenue commission to inquire about any possible refund of St. Louis local tax withheld after you moved to MS. TurboTax does not have the forms to prepare a local St. Louis return.
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