Typically speaking, you will have 60 days from when you made your move until the day you enroll in your new coverage (or make a change in your plan). If you were moved to the District of Columbia by an employer, then your healthcare change should be near immediate.
The Medicaid program is handled through a program called DC Healthcare Alliance which provides treatments and services like hospital care, cancer screenings, blood pressure treatments, preventative care, and more if your income is below a certain limit.