It is a very long picket fence so I expect you would like to plant a mix of different plants. It is not always a good idea to plant a monoculture (many plants, all the same) as it creates an environment where insects and diseases can destroy the entire planting, under the right conditions.
We understand why you would like an evergreen, but there are none that are native to your area that will thrive in your conditions of sun and dry soil. If you do a Combination Search of our Native Plant Database for DC and select: shrub/sun/dry soil and both 3-6ft & 6-12 ft sizes, it generates a list of 12 suitable shrubs, none of which are evergreen. If you repeat the search selecting trees 6-12 ft, you get one more, Ilex verticillata (Common winterberry), which is actually a very large shrub anyhow.
So we recommend you consider a deciduous planting instead. It will be interesting and attractive, easy to maintain and will offer flowers for you and habitat for birds (food and cover). A few from that list we think you should consider are:
Physocarpus opulifolius (Atlantic ninebark) which is "tough as nails" and has attractive red seeds after the flower is spent
Rhus aromatica (Fragrant sumac) which produces bright red fuzzy berries on the female plants and has great fall color
Rosa setigera (Climbing prairie rose) which will produce a huge amount of blossoms when attached horizontally along the fence and produces bright red rose hips that birds love
Vaccinium corymbosum (Highbush blueberry) that will give you blueberries for your cereal if you get them before the birds do and also has wonderful red fall foliage
Viburnum acerifolium (Mapleleaf viburnum) which produces red to blue/black fruit and turns pink/burgundy in the fall
and of course
Ilex verticillata (Common winterberry) whose females produce red berries that persist along the bare arching stems well into winter and are great for holiday arrangements
Here are some photos from our Image Gallery: