According to "Landscaping with Native Plants of Texas and the Southwest" by George O. Miller, silktassel, Garrya ovata, aka Garrya lindheimer, might fill your needs. The book says, "You couldn't ask for much more in a landscape plant. Silktassel is evergreen, fast growing, densely foliated, and drought and shade tolerant. The only thing lacking here is showy flowers and fruit, but I guess you can't have everything. When used alone as a specimen plant, silktassel develops a full, symmetrical shape, growing as large as 10 to 15 feet. The lustrous green leaves contribute year-round color to a mixed planting with deciduous species. It makes an ideal visual screen or privacy hedge, or you can prune it into a medium-sized foundation or boundary hedge. A 6-inch seedling in my yard grew into a 10-foot bushy shrub in four years." That said, I have no experience with it myself.
Another possibility would be wax myrtle, which probably would want more water. Mine will take standing water. It survived the drought last year, but lost a large trunk. It's multi-trunked, and spreads quite a bit. After I put these in the yard and they got good sized, ruby-crowned kinglets and yellow-rumped warblers started wintering in the yard. There are male and female plants, and to get the berries that attract these birds, you need a female plant, with a male pollinator close. I read, on a competing forum, that there has been a problem with them farther north, as in your area, with unexplained die-back, after they are several years old. That is the only place I've seen that information, but I would check with the extension agent in your area before investing money in it.
There is also a dwarf wax myrtle, that gets 6 feet tall and spreads even more prolifically, from what I've read. I tried it, but it didn't survive. I think there was cotton root rot in the soil where I put it, though.
Yaupon holly is slower growing that you want, I think.
