In the scenario you present you indeed are a statuatory resident of Washington DC (see below) and domiciled resident of NY. You are a dual resident. So for your original question, which tax return to work first, it doesn't matter. There is not going to be any credit for taxes paid to another state on either return because both returns will be resident returns. You will be paying taxes to both states (DC/NY) on all income from all sources. That would also mean including the bank interest on both returns.
You will most likely not want to change your domicile due to the loss of dependent/education benefits to you and your parents on their returns.
Since the income was likely earned while physically in DC, I suggest in the "My info" section you indicate NY is your resident state and that you earned income in another state and select "DC" from the dropdown menu. Enter all the income and deductions in the federal interview and when you get to the DC return, you will be asked to confirm the correct residency. You will "change residency" to "Resident."
(42) “Resident” means an individual domiciled in the District at any time during the taxable year, and every other individual who maintains a place of abode within the District for an aggregate of 183 days or more during the taxable year, whether or not the individual is domiciled in the District, excluding any elective officer of the government of the United States or any employee on the staff of an elected official in the legislative branch of the government of the United States if the employee is a bona fide resident of the state of residence of the elected officer, or any officer of the executive branch of the government whose appointment was made by the President of the United States and subject to confirmation by the Senate of the United States and whose tenure of office is at the pleasure of the President of the United States, or any Justice of the Supreme Court of the United States, unless the officer, employee, or justice is domiciled within the District at any time during the taxable year. In determining whether an individual is a resident, an individual’s absence from the District for temporary or transitory purposes shall not be regarded as changing his domicile or place of abode.
See page 7 here as well for who must file as resident in DC.
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