Ask Mr. Smarty Plants

Since you state that your soil is sodic and your water has high sodium and sulphur content, I assume you have had it tested.  If you haven't, you should submit a sample to the Soil, Water and Forage Testing Laboratory of Texas A&M AgriLife Extension Service.  

My search for plants that are tolerant of sodic soils resulted in only one recommendation.  Salinity, Sodicity and Flooding Tolerance of Selected Plant Species of the Northern Cheyenne Reservation from University of Montana has data that shows only one plant, Glycyrrhiza lepidota (American licorice), as being tolerant of sodic soils.  It isn't a particularly attractive plant and may not grow naturally in Denton County.   The nearest counties to Denton County that it is shown as occurring on the USDA Plants Database distribution map are Wichita and Ellis Counties.

In the Colorado State University Extension Service article, Managing Sodic Soils, there are suggestions for amending the soil by substituting calcium for sodium.  One of the ways they suggest doing this is by adding a sulphur product to make the soil acidic in order to dissolve calcium carbonate from limestone in the soil.  Since you already think you have a high sulphur content it seems like your soil should already be somewhat acidic—but, perhaps your soil doesn't have an excess of limestone in it.  Another way to acidify the soil and possibly dissolve calcium carbonate from limestone in the soil is to add organic matter that will oxidize and create carbonic acid. Another suggestion is to add gypsum (calcium sulfate), but if your soil is already saline this will probably make it worse.  Unfortunately, adding soil amendments to a large area is not going to be an easy task.  The article has instructions for amounts of different products and how to apply them.

The Colorado State University Extension Service article observes that sodic soils are poorly drained and tend to crust, but that they often respond well to good irrigation water and practices.  One way to alleviate the poor drainage would be to dig narrow drain trenches on the downslope from your plants.  Enhanced drainage will help flush out cations (e.g., sodium).  Also, instead of doing supplemental watering from your well, collect rain water to water to use in supplemental applications.  Finally, you might consider building raised beds, importing good soils and growing plants in them.

It would be a good idea to contact the Denton County Office of the Texas A&M AgriLife Extension Service to see if they suggestions for your soil problem.   There are probably other people in Denton County with similar problems.

You don't mention problems with salinity, but here are a few choices for plants that are at least moderately salt tolerant.

Amorpha fruticosa (Indigo bush)

Gaillardia pulchella (Firewheel)

Asclepias tuberosa (Butterflyweed)

Hesperaloe parviflora (Red yucca)

Ilex vomitoria (Yaupon)  There are dwarf versions, e.g., 'Nana'.

For other plants for your area visit our Texas–North Central Recommended page for a list of commercially available plants for landscaping.  You can use the NARROW YOUR SEARCH option to add your preferred criteria.   Unfortunately, there isn't a spot to select for soils with high sodium content!