I have a tricky case for that I need your urgent assistance. Let me brief my case below...

Me & My wife both stayed & worked in NJ till Oct middle in 2013. From Oct middle we both relocated to Delaware as my wife changed job. So from oct middle till rest of 2013 my wife was staying & working in Delaware but I was staying in Delaware but working in NJ. So ideally I have 1 W2 for all my income form NJ only & my wife have 2 W2s 1 for income from NJ & 1 for income from DE. Now i am confused how should I file my taxes for 2013. My questions are as below.

1) Should we file jointly or separately?
2) If we file jointly then how should I file NJ tax. As I understood after reading NJ tax rules seems like they say to file both part year resident tax & part year non resident tax as I have earning from NJ after Oct middle while I started staying in DE. Please suggest.
3) If we file jointly then how should I file DE tax. Should i file as part year nonresident tax (i.e. 200-02 form) in DE? Please suggest.
4) Please suggest in which case I need to file tax credit for income from other state if there is any double taxation scenario in my case.

I am too much confused & looking for you urgent help on this.
Thanks.