First, you will want to do the nonresident state first, so do NY first. This will calculate the NT tax first.
Then do your DC return as a resident. The taxes paid to other state will appear as "Out of State tax credit", please see page 14 of the DC Booklet.
Note that the instructions list this as a nonrefundable credit, which means that the credit will be applied to your assessed tax but no more. If some cases, taxpayers may not see this credit, because either they did not have much tax assessed in the resident state or other credits gobbled up the assessed tax or whatever.
But try doing the states in the order I suggested and see how it works out.
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