It is normal that you pay tax to the state where you live and to the state(s) where you work. In this case "you" is both of you.
The way that many (not all) states deal with the issue of double taxation is that any taxes paid on income in one state is used as a tax credit in the other state(s). In your case, Delaware starts out with the federal income (more or less), then calculates the state tax.
Then, when you do the NC tax return, TurboTax will complete SC form D-400TC. The first part of this form calculates the tax paid to other states and applies that as a credit to NC taxes.
In short, you will pay state tax on your federal income somewhere but it just may not be to your home state.
Reminder, to get this to work, be sure to enter your non-resident state first into TurboTax, then enter your home state, so that TurboTax can calculate the nonresident state tax first and carry it to the resident state.
"Do we need to file separately?" - we do not have nearly enough information to be able to answer that. I would say that the only way that you can be sure is to run the tax return jointly and then separately to see the difference.
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