You filled out the forms wrong. NJ allows a credit for tax paid to NY on income also taxed by NJ, and NY taxes are usually higher than NJ taxes, so the credit is usually as much as the NJ tax, so you don't pay any extra if you figure the credit correctly.
If you are absent from NJ only to attend school, then you can't change your residency. Once you are a permanently resident of a state, you remain a permanent resident until you move permanently -- meaning that you are gone for good, not just until you finish school.
Also, if you change to a resident of New York City, then you have to pay City income tax, in addition to state income tax, so it will cost you more, not less.