Can I file my Virginia and New Jersey tax returns separately?

Yes, you will file a New Jersey and Virginia return.  Each state has its own requirements.  TurboTax will guide you through the entries, but if you have questions when completing the returns, you can post here.

New Jersey does not have separate tax returns for part-year residents- either Form NJ‐1040 or Form NJ‐1040NR (or both) is required, depending on your residency status during the year, and show the income you received during the period of time covered by the return. 

A person who lives in Virginia, or maintains a place of abode in Virginia, for more than 183 days during the year is considered a Virginia resident for income tax purposes.