Should state taxes be the same as LCA location?

I’m on H1B visa since November 2013. My employer is headquartered in state A, but my LCA was approved for another state B. I have been working in state B, but my paystubs were generated with state A taxes. Should I file two W2 forms to reimburse state A taxes and file state B taxes for 2013. Also, my paystubs dates don’t match the paycheck dates - I was being paid regularly, but paystubs couldn’t be generated for the first two months on pay days since I didn’t have an SSN. As soon as I got my SSN, my employer generated paystubs of the past 2 months - but their date is pushed back to 2014 - Does that mean I don’t need to file 2013 taxes (since all paystubs are dated 2014) or am I required to file 2013 taxes (since I was technically working over November and December, being paid on Dec 1 for Nov and so on, but paystubs are all dated 2014)

Thanks a lot