I get cold easily, but I'm still comfortable off the coast in a 3mm full as long as I wear a hood or beanie. I've seen people dive in shorties, but the full suit is good for protection, just in case you accidentally bump into something (and all of the old pros wear full suits). You should avoid touching wrecks or reefs, but just in case... Thin gloves are good because you will often be holding on to the line for ascents/descents in current, and sometimes the line has sharp things growing on it. Plus touching a wreck with bare hands is never a good idea.
As compared to Cozumel, you'll notice bigger waves on the surface, which makes getting in/out of the water more challenging, so listen carefully to the boat crew for specific instructions on that, and be prepared to work hard to get up the ladder.
You'll also notice lower visibility, so you have to stay close to your buddy and close to the wreck. The shallower, inshore wrecks usually have pretty low visibility and less current, while the wrecks in the gulf stream usually have better visibility and more current. I thought the current in the gulf stream was comparable to Cozumel drift diving, although on a wreck dive you have to return to the anchor point, so you have to stay close to the wreck to stay out of the current, and swim into the current first if possible so you can get back easily. You also have to pay attention to where you are so you can find your way back to your own anchor line. Most people also carry an SMB with them just in case they can't get back to the boat.
Another difference is that in Cozumel, they usually hold your hand and baby you a lot. On the NC dive boats, you're expected to haul and set up your own gear, you're expected to plan and execute your own dive (but you can definitely get help with the planning), and you won't have a DM unless you pay extra for one. We hired a DM on our first NC dive, and we were glad we did.
The surface interval will be very windy and might be cold and wet, so a boat coat or a wind-and-water-resistant jacket are necessities. I'd bring a change of clothes in a dry bag too. The boat rides are significantly longer, so bring some good snacks, and take seasickness meds before you leave if you are even a little worried about seasickness.
God, that sounds awful. All of that aside, the diving off of NC is fantastic. I'm sure you'll love it!