Generally speaking, if your permanent residence is in DC (your domicile) and you temporarily lived and worked in New York state, then you will owe NY state tax on the money you earned while living in NY (and NYC taxes too, if you lived in the City). You will owe DC taxes on all your income, but DC will give you a credit for the tax you paid in NY.
In Turbotax, you would file a NY non-resident return and a DC resident return. Prepare the NY non-resident return first so the tax credit flows correctly. Turbotax can handle this situation, or you may upgrade to the Turbotax Live Online version when it is available, or you can go to a tax pro elsewhere.