welcome!!!
Nice to see another SW'er out here. I grew up in the LA area but spent lots of my youth exploring the deserts of the SW as well as the mountains (if you are a lover of the mountains you will miss them most of all out here, especially in jacksonville) I explored much of AZ and my dad is building his retirement home in Sedona.
That said NC is a great state
There is beach driving not too far from you. OBX, FT fisher, Carolina beach come to mind. But it is beach driving, not too technical but also not very busy on the beaches unless its a holiday weekend so it is nice to use them to get off the main road, away from people and down near the water.
To really get away (what you would be used to back home in the desert) then Portsmouth island would be your best bet. You have to take a ferry to get there and it is just sand and marsh driving all over the island.
There are two designated OHV areas in the state. Uwharrie (our club goes here 3-4 times a year) and Brown MT. Brown MT for 4wd vehicles is basically a long dirt road loop with a few areas that could be "challenging for a stock height rig" but mostly just a dirt road. Great views and great camping but it is a atv/dirt bike part mostly.
Uwharrie has about 16 miles or so of trails for us to explore and our club also sponsors a trail there and does work days for it.
There are a few other places to get off the beaten track, not technical at all though and can easily be done in a stock LC.
Hurricane Creek is another good "expo" style trip as well as black balsam.
There are other places to wheel in other states scattered out there.
The hills of NC are nice to go to in the summer months to escape the heat. They are nice to drive through.
You won't find anywhere near as many places to wheel out here but you are in luck because the greatest LC club in the land resides in NC and you have found them. We also do at least 1 trip a month with regional meetings. There are a few members out by the coast also. Adam @custyota is one of them.
Check out our club calendar for upcoming events.
https://forum.ih8mud.com/threads/2015-onsc-calendar.834810/
As far as living in NC on the east coast. It will be a culture shock at first, just like moving anywhere far from home. The first year my wife and I were here we didn't know anyone outside of 2 family members so we spent our free time just driving to different parts of the state and walking around. Its a great way to see the state and get out. Getting out and exploring will help with the culture shock you will get going from AZ to NC
PS if you like craft beer NC is one of the up and coming states, its starting to creep towards my top 5. There are some top notch breweries here.