In general, you report and pay income tax on all your world-wide income to the state of your permanent residence. In this case, New Jersey.
You would only pay Delaware income tax on what is considered Delaware-sourced income. Wages you earn while living in NJ will not be considered DE-source even if the company is based in DE, because what counts is where you live or work at the time. (There are some exceptions but DE is not one of them.). However, if you physically work in DE for even one day (for example, 1 day a month for meetings, or 2 weeks a year for training), that part of your income will be considered Delaware wages (whether you were staying in a house, hotel, or whatever).
You should start by having the company only withhold NJ income tax, not DE state income tax. Then, if you did have DE-sourced income, you would file a DE non-resident return to report those wages and pay the tax. (You will manually allocate your wages to DE based on how many days of the year you worked inside DE compared to the total working days in the year. Turbotax will help with this but Turbotax doesn't know how many days you worked in DE.). You then can claim a credit on your NJ full year resident tax return for taxes you paid in another state for the same income.
