Do I file a Part-Year resident return for Virginia and New York, and file Non-resident for Maryland?

It depends.  This is admittedly a little tricky, because Virginia and Maryland are reciprocal states.  Normally that arrangement makes filing state returns easier, but not in this case.  This is because by statute you fit the description of a nonresident for Maryland, and more so since you correctly state that TurboTax (and many states) do not allow for three part-year returns to be filed (when both a move-in and a move-out of the state occur in the same calendar year).  Under "normal" situations, when you file a part-year resident and a nonresident return, your resident state gives you a credit for the amount of income you earned while working in the nonresident state, which is accurate and not that difficult to prepare in TurboTax.  

But, because VA and MD are reciprocal states, they will not allow you to claim a credit in this way.  You are simply taxed to your resident state, and you claim a reimbursement of mistaken withholdings (both VA and MD have special returns for this) to get the nonresident state to return the state tax to that state.  For your situation, this is probably what I would recommend:  file Virginia and NY part-year returns, and treat your MD residency as if you were still living in Virginia at the time, so that you may file a mistaken withholdings return to Maryland and get back Maryland taxes (but then VA taxes that portion of income).   

Admittedly, that is not 100% accurate to your situation, but it most faithfully holds to the spirit of state tax law without making your return overly complicated.  Here is a screenshot to assist you to set up the Maryland return should you go this route of filing: