Ask Mr. Smarty Plants

The Houston Chapter of the Native Plant Society publishes a Native Plant Guide on their website with a lot information about plants for the Houston area.  One of the articles in that guide, "Native Shrubs That Thrive in Poorly Drained Black Gumbo Soil" has several suggestions:

Euonymus americanus (American strawberry-bush) is described in our database as being an "airy" shrub so this may, or may not, provide you with enough privacy.  Here are more photos and information from Carolinanature.com and Vanderbilt University.

Morella cerifera (Wax myrtle) seems like the ideal shrub since it is evergreen and fast-growing.  The wax covering its berries and leaves is flammable, however, so I understand your concern.  It would require a source of combustion (e.g., a match, a cigarette), though, to set it aflame—they don't experience spontaneous combustion.  Here are more photos and information from Duke University.

Cephalanthus occidentalis (Common buttonbush) would do very well when the soil is wet, but not very well if the soil becomes very dry.  It has attractive ball-shaped flowers and fruits.  Here are more photos and information from TAMU Aggie-Horticulture.

Itea virginica (Virginia sweetspire) does well in moist soils, but usually grows to only 3 to 6 feet although it can reach 8 feet.  It is also described as being somewhat scraggly as a single plant.   It does have attractive flowers.  Here are more photos and information from North Carolina State University.

Ilex vomitoria (Yaupon)—evergreen—and Ilex decidua (Possumhaw)—deciduous—both tolerate wet soils and female plants have beautiful red berries that carry over through the winter and provide food for various bird species.  Her is more information from TAMU Aggie-Horticulture for Yaupon and  Possumhaw.  The growth rate for Yaupon is described as moderate and that for Possumhaw as slow by the Texas Forest Service Tree Planting Guide, but both can usually be found as moderately large plants at nurseries.

Cyrilla racemiflora (Swamp titi) would do very well in your soggy area, but not likely to do well in very dry soil.  Here are more photos and information from TAMU Aggie-Horticulture.

Sabal minor (Dwarf palmetto) is a palm and not technically a shrub or a tree.  It could, however, be ideal for your purposes.  It will do well in swampy areas and in drier areas as well.  It is evergreen and easily can grow to six or more feet tall.  They are somewhat slow in growing, however.  Here is more information from North Carolina State University.

Viburnum nudum (Possumhaw viburnum) grows 12 to 20 feet tall and has showy white flowers and good fall color.  Here are more photos and information from Floridata.

The information I found for Chionanthus virginicus (White fringetree) does indicate that it would do well in your location.   The North Carolina State University page does indicate that its growth rate is slow to moderate.

You might consider modifying the area by laying a French drain.  This would give you more choices for the type of shrub/tree you could expect to thrive there.