I imagine that you already suspect that your situation is difficult and your choices limited.
As a matter of fact, when you consider all the variables, the only evergreen that will grow in dry shade to a height of more than six feet and is native to your area is Ilex vomitoria (Yaupon). If you are willing to give the plants supplemental water you can add Ilex glabra (Inkberry) and Morella cerifera (Wax myrtle) to the list. They are all plants that are commonly used for screening and could do the job for you. It is unlikely that they will grow quickly, though, as life is not easy in the soil between the roots of 75 year old trees.
It is impossible for us to make a recommendation about a situation we cannot actually see, but you may need to look at alternative ways to create the privacy you need. A landscape designer may have suggestions of how to combine sections of fence and plant groupings in strategic locations to avoid planting directly below those grand old trees.