Education

No. You will not need to file in MA and maybe your dad can claim you as a dependent on his return assuming you were a full-time student Jan-May.  For you to be a dependent on your dad's return you need to pass the following tests to be a qualifying child.  Here is a link to IRS Publication 501 for more detailed information.  (See table 5 and Qualifying Child)

• The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, or stepsister, or a descendant of any of them.  YES
• The child must be (a) under age 19 at the end of the year and younger than you (or your spouse if filing jointly); (b) under age 24 at the end of the year, a student, and younger than you (or your spouse if filing jointly); or (c) any age if permanently and totally disabled.  Assuming Yes.  Definition of student is:  full-time according to your school program for any part of 5 months in the tax year.  
• The child must have lived with you for more than half of the year. Maybe (time away for school or on vacation counts as time living with parents) 
• The child must not have provided more than half of the child’s own support for the year. Probably. At least into August parent's only support. 
• The child must not be filing a joint return for the year (unless that joint return is filed only to claim a refund of withheld income tax or estimated tax paid). Yes

Assuming all the tests are passed.  In the "My Info" section on your return, you would indicate that "you could be claimed as a dependent on someone's return."  You would select that your resident state on 31 Dec was NY.  Select "Yes" you lived in another state and indicate the date moved to NY (maybe when you got the NY job).  

This will generate part-year returns for NY and MA.  In the NY return you will allocate all income/deductions to NY.  There will not be any income allocated on the MA return so you will not need to file that one.