I am trying to figure out if we should file married jointly or married separately on both the federal and state returns. I live in Philadelphia and worked for half the year in Philadelphia and the other half in New Jersey. My husband works and lives in New Jersey. Should we file married jointly or married separately on our federal and state returns? Or is it possible to file married jointly for federal and married separate for state?

Usually it is cheaper to file jointly.
There is only one way to find out - calculate both ways.

JoyPulv is correct; the return needs to be modeled BOTH ways to be sure.

However, most tax softwares have a JOINT vs SEPARATE tool which does a quick check to see if filing separately makes more sense than filing jointly.

Such tools MAY be wrong, but it can give an indicator to do the modeling.