you need to consider that going from NYC to NC even is a car is torture. if you are planning to slab it you will be relatively miserable depending what kind of temperment you have. It will be a boring, windblasting, mind numbing ride.
That said.. I'd probably do it rather than pay to ship the bike.
If google maps says it takes 6-8 hrs, add at least 2 if you're on a bike, gas stops, piss stops, food stops, dead phone etc. Wear a camelback or similar, stay hydrated. have some snacks in a tank bag or pocket, bring a USB power bank for your phone.
So not knowing your timeframe/finances/how much you plan to take with.. I think I'd say tow the bike. If that's not an option it's doable but it might be miserable.
If you want to do it without it being complete misery make it a minimum of two days. If you don't mind traffic and riding slab, take your time on the highways, spend a night in DC or Richmond. If you want to make a better ride of it then head west through PA and come down rt 15 to rt 340, you can even get a taste of skyline drive or the blue ridge parkway (keep in mind it takes a lot of time to go any distance on these twisty roads, but if you have to go from NYC to NC what better reason). If you have time, you could make an awesome trip of it.
check out https://trips.furkot.com/
if you want to do some trip planning.
I've got a couch you can crash on in Frederick, MD if you need it, just PM me.