I moved in March from NY to AZ how do I change my state of residency?

Moving somewhere to work and moving somewhere to live are not the same. If you are truly moving and settling in, you will change your license, voting, social scene and file a part year tax return showing your move dates.

Legally, for NY, the DMV is the only big consideration. See Moving out of New York State - NYS DMV.

In the Personal Info section of TT, enter AZ as your State of Residence, and indicate that you also lived in NY, Enter the date you began living in AZ.  As @AmyC stated, you'll be filing a part-year resident return in each of the two states.

How do I file a part-year state return? (intuit.com)