Yes, you will have to file state taxes to both NY and to NJ. Most state tax returns begin with the AGI from the federal return. This would carry over from the federal return. The full amount would be reported to both states. Prepare the NY non-resident return first and then the NJ resident return. In order to get the correct amount of credit, also prepare the non-resident state first.
As far as paying taxes to both states on every paycheck, that is something you will need to take up with your employer. To avoid paying taxes to NJ you would want to have taxes withheld from your paycheck. If your employer will not take out for both states, then you could make estimated payments to NJ so that you have taxes paid in and would not have to pay at tax time.
