Ask Mr. Smarty Plants

First of all, you need turn under or cut the rye grass before it goes to seed or you are going to have lots more rye grass next year.  Next, sedges are definitely good choices, but I think your best bet is to stick to the species that are native to the area:  

Carex perdentata (Meadow sedge)

Carex planostachys (Cedar sedge)

Carex texensis (Texas sedge)

Carex cherokeensis (Cherokee sedge)

Several of these sedges should be available for sale at the Wildflower Center Spring Plant Sale (Friday, April 13 for "Members only"; Saturday and Sunday, April 14-15 for general public).  The way to establish the sedges is by small plants that can be inserted in the soil within the rye grass. They will grow and spread from the small plants.

Both of the following groundcovers should also be available as small plants at the Plant Sale.  You might consider using a combination of sedges and these two to form your groundcover:

Calyptocarpus vialis (Straggler daisy) will grow in the sun but loves to grow in the shade and part shade.  

Phyla nodiflora (Texas frogfruit) grows well in sun and part shade.

Both plants are semi-evergreen depending on the severity of the winter weather.

Since your yard has full sun, it is a great candidate for the Habiturf lawn developed by the Wildflower Center.  It will go dormant in a cold winter, but become green again with spring warmth when the rains fall again.  Once established it requires little, or no, watering or mowing.