In answer to your questions,
You can take your children out of state, but your spouse has the right to sue you for emergency custody bringing the children back to NC. Generally it is not advisable to move the children out of state without an agreement.
Dating before a separation agreement and before divorce impacts both alimony as well as potentially alienation of affections. It is advisable that you consult with an attorney directly, to determine how much of an impact dating can have in your situation.
Lisa M. Angel
Board Certified Family Law Specialist
The Rosen Law Firm
4101 Lake Boone Trail, Suite 200
Raleigh, NC 27607
(919) 781-1741 direct voice
(919) 256-1660 direct fax
(919) 787-6668 main voice
(919) 787-6361 main fax
The response posted above is based upon the limited factual information made available and is not intended as a full and complete response to the question. The only reliable manner to obtain complete and adequate legal advice is to consult with an attorney, fully explain your situation, and allow the attorney sufficient opportunity to research the applicable law and facts required to render an accurate opinion. The basic information provided above is intended as a public service but a full discussion with an attorney should be undertaken before taking any action.