I live in Central Texas and have encountered the same problem. For a living barrier, I planted vitex and they have done quite well. While the plant has shown up on the Texas invasive list, it's never been a problem for me. Others in the area use xylosma (a shrub from China) and elaeagnus. Both of these grow quickly and provide privacy. However, my favorite green screen is the Winter Honeysuckle because it stays evergreen and blooms early so provides nectar for the first of the migrating butterflies.
Fall is the perfect time to plant!
Hope you enjoy your new home.
