In the Personal Info section of TT, you should enter CT as your State of Residence. You should indicate that you also lived in another state (NJ), and you should enter the date you began lived in CT.
You will file a part-year resident return in each of the two states and allocate your income accordingly.
In the Mailing Address section of Personal Info. you should enter the mailing address you want the IRS to use if they have to mail you a letter or a check.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.