DMK,
I have been a sales rep "down east" for the past 26 years, and travel the area extensively. While there is not an overabundance of public ranges there are enough to satisfy the need. You have been linked to the Dare County (Outer Banks) range above, moving south there is a nice range with a small but well stocked gun shop near Havelock (B&R) that is also referenced above.
A little further inland there is the Pitt County Wildlife Club, in Pitt County, just off of NC43 west of Greenville (they have a website) - the PCWC is open to members and their guests. 13 miles south of Greenville (on NC43) is a new public range operated by Paradise Outdoors that is open 7 days per week offering annual and daily memberships.
My favorite public range near the coast is Flatwoods Outfitters in Hubert NC (just off of NC24 between Jacksonville and Swansboro) - it is owned and operated by Mike Stroff, Lt Col, USMC (RET) - terrific gun shop and range. In Rocky Mount there is the Old Hickory Gun Club (I am a member of that club) it offers both daily shooting ($10.00 per day) or annual memberships ($100.00 per year). OHGC is open WED & FRI afternoons, Saturday and Sunday afternoons - it is primarily a Skeet shooting club, but has a nice rifle/pistol range and a very good CAS club/range.
There is a public range near Southport, I have seen the sign but have never turned in and cannot remember the name of the range; same story between Spring Lake and Lillington on NC210 - nice sign but I have never turned in to look. The very old and established Buccaneer Gun Club is off of US 17, south of Wilmington, I have not been there but a friend of mine goes there occasionally to shoot in skeet tournaments. There are more (one near Burgaw, off of US117 that I have forgotten the name of also). If you search you will find a few more. The gun shops often can lead you to a range.
As to permitting, it depends on counties (as it does in the rest of the state). My sheriff will issue permits on the spot at gunshows and overnight in his office. I know Wake (Raleigh) can take weeks and I have heard that Durham and Orange (Chapel Hill) can take months. A friend of mine from Washington (Beaufort) obtained permits there a few weeks ago in 4 days. Another friend in Fayetteville always has plenty of permits for gunshows, trading, etc. CCW, as you know, is "shall issue" in NC, and I have heard of no Sheriffs in our area (central part of eastern NC) attempting to subvert the process, but I don't know what happens in each and everyone of the 40 counties on and east of I-95.
Odessa
http://www.pittcountywildlifeclub.20fr.com/index.html
www.oldhickorygunclub.com
www.paradisehuntingpreserve.com
http://www.ncrpa.org/ranges.htm
http://www.n-ssa.org/CAROLINA/MapJacksonville.jpg