You cannot be double=taxed on state taxes. DC is not a state but has a district income tax that works like most other states. Your employer will probably take out just NY taxes. Or may take out DC taxes, or might now do any state withholding if you apply to exempt from state withholding.
If there is no state/district withholding, then you can pay estimated taxes to NY state. If taxes are withheld for DC, file a NY and a DC tax return. You will have the DC tax refunded and pay for NY state.
