State tax filing
For New Jersey, you will have to report the December income (it will be a nonresident return). See the screens below to see how to allocate the NJ resident and then the NJ nonresident income. (The screen shots are from the desktop version so they may be a bit different if you use online but will be essentially the same).
For Delaware, you should file as Part-Year Residents electing to file a non-resident return - This option may be advantageous if, during the period of non-residency, you had any income from other states or sources outside of Delaware. You will report zero W2 wages (for the NJ work) and enter the December dividends etc as DE income.
TurboTax will guide you, but if as you are working through you need help or have questions, please post!
