I would have mentioned "Agarista populifolia (Florida Leucothe), Illicium floridanum (Florida Anise Tree) ".

Allie,
Plant your self some sweet box shrub there.
They will be fine with a little sun in winter, and they love
shade in the summer.
They will need to be watered tho, mabey with a drip hose in the heat till they get established.
The smell of this shrub is very good, smells like sweet vanilla, and it blooms in Feb thru April.
You can just prune it down eventually if you want, although they grow pretty rounded when they get big.
You can buy them in any big box store, in the spring they usually have them.
They don't cost that much either, look online to see how far apart you should plant them.

I would go with illicium parviflorum. It is very hardy and gets to 10ft. but will need supplemental water. Tiny yellow flowers and fragrant leaves are nice.

other possibilities could be english laurel, southern wax myrtle if enough sunlight to keep the foliage dense.
aucuba japonica is a tough shrub for heavy shade. comes in green and variegated leaf colors.
the viburnum you mention is a nice plant, too.
on a side note, my sarcococca (sweet box) is not yet 2.5' tall after 8 years. maybe she is not happy in the dry shade location where i planted her. and i still can't detect the fragrance reported for this plant.

allielee, behind the tree trunks, is there a pitch down to your
neighbor's lower level (behind the fence) ?
(concern for competition for water by the big trees and any sharp drainage from topo change)
Is that your driveway this side of the rubble rocks and are you planning to plant behind the rocks among the oaks or closer to the fence line?
Is that fence yours or theirs?
Aspects to consider before laying out the $$ for shrub evergreens and trying to do an install amid large tree roots.

Thanks everyone for the advice. Any more suggestions are always appreciated!

Dottie: thanks for bringing up those concerns. The fence is mine. There's no pitch down to my neighbour's house, although there is a slope from my neighbour's front yard going all the way through our yard. There's no driveway there; it's just all garden. I was also concerned about the trees hogging water. I'm going to cut down the little one by the fence, but I'm not planning on doing the others. Do you think I could just water them well the first couple of years? I hope to plant the hedge up against the fence.

Here is a link to a great north carolina site for native plants, If you click on the Shrubs part you will find all the sun/shade needs and water needs.

http://www.ncwildflower.org/natives/recommend.htm

I am a big fan of a mixed shrub border, and it looks like you have many stones that you could intermingle in there. Instead of trying to just screen the unpleasant view, create your own very pleasant view with a few kinds of native or low maintenance other shrubs of varying height and foliage. Native shrubs are very easy to maintain, just find the ones with the heights you need. You can put taller in back and a few lower in front, looking very natural and beautiful, and another benefit is you will attract birds and provide shelter and food for them. Lovely all around!

Hard to suggest not knowing exactly how much sunlight this spot gets under the trees.
I will say in reminder that you consider the mature width/bulk of whatever shrubs you choose and plant at least 2-3 feet away from the fence for better air flow for the health of the shrub.
Use well rotted compost to amend the soil rather than commercial fertilizer because you don't want to be attracting more feeder roots of the big trees to sudden nutrients.
Whatever you plant needs the first couple years to establish its roots among the tree root area so years 3-4 after install you should see more growth.
I'd honestly leave off the 'mulch' so it's easier for you to check for water needs and also spot if any voles decide to take up residence in the soils you've loosened for planting.

I have curly ligustrum in a very shady spot against the house and shaded from morning sun by a broad vitex tree.
The 'curly' has very dark leaves and interesting form, more tall than broad in shade. Might make a nice mix with leucothoe fronted by pieris and tuck in a few daphne odora
and perhaps a mahonia or two for interesting yellow flowers and blue berries in the shade.

Put a couple Southern Magnolias there, they'll hide all of that really well and they can handle shade.