Leucophyllum frutescens (Texas cenizo) makes an excellent hedge and produces beautiful pink/purple blossoms year round in response to rain.
Morella cerifera (wax myrtle) has leaves that produce a pleasant scent when crushed. Its berries attract birds and other wildlife.
Rhus virens (evergreen sumac) is not a true evergreen since it loses its leaves in the spring, but it replaces them immediately. It also has berries used by wildlife.
These next ones are slower growing but otherwise meet your criteria.
Ilex vomitoria (yaupon) has striking red berries that attract wildlife.
Sophora secundiflora (mountain laurel) is not fast growing, but has beautiful, sweet smelling flowers and would florm a dense barrier.
Garrya ovata ssp. lindheimeri (Lindheimer's silktassel) is an interesting evergreen shrub, but may be hard to find in nurseries.
Finally, one other suggestion for a fast-growing, dense fence would be the much-maligned Juniperus ashei (Ashe's juniper or cedar) tree. Its advantages are that it is fast growing and can form a dense cover. Its disadvantages are that its pollen causes allergies (cedar fever) and it is considered invasive. If you plant all female trees, you shouldn't have a concern about the pollen. It is considered invasive because it tends to take over pastures forming dense thickets, but used as hedge in your yard you could control its spread.
