Yes, do the two states in the order you suggest, then look at line 1 of Schedule U on your DC return and see if your New York state taxes appear there as a credit in DC.

...........but, 100%credit ?....perhaps not.  DC would allow a credit of "up-to" what DC would have charged for the same income.  So if NY is taxing at 6.2% and DC is taxing at 4%, DC would only allow a 4% credit.

But that really depends on what the total double-taxed portion of your income is,  because DC and NY have progressive rates:  NY is 6.21% to 8.82%, and DC is 4% to 8.5% over differing income ranges.....which is why the software is needed to puzzle it all out.