campanulas that grow low and creep about. I'm trying to recall the exact name but it's pretty and easily controlled.

thrifts -- in pinks and white variety. So pretty. Again easily controlled.

a mix of succulents and there are so many to choose from. Your zone is perfect for these plants.

black mondo grass.

aranaria

I am in such a different zone than you that you will have to check these out:
Veronica Georgia Blue
Cranberry makes a great ground cover in ordinary garden soil.
Dianthus Greystone
Perhaps some of the non climbing clematis, depending on whether you have slugs or snails.

All play well with others in my gardens.

What about Verbena x hybrida?
I find these so useful for edging plants, or meandering through other plants. They flower almost year round. They are shallow rooted and easy to pull out if necessary.
They come in lots of colours, so it is easy to find ones that go well with your other plants.

Here is a pink one with a purple sage.

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....and a purple one the other side of the sage.

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(Sorry about the hose!)

Here is a lighter pink one, trailing down a bank, behind the pond.

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This is one of the Tapien series of verbena, the ones with the finer leaf, growing with Calamintha nepetoides, not yet in flower.

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There is also Dichondra Silver Falls. This works well as a ground cover.
Here it is with petunias growing through it.

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Hope this helps.
Daisy

I use creeping jenny as a ground cover,doesn't flower,but stays nice and green year round.
Kathi

I use horse herb. It stays low.

Here is a link that might be useful: tx horse herb

perennial geraniums.
There are dozens to choose from and I'm sure a few can tolerate Texas heat. Find the website geraniaceae.com There is a sections labled "groundcovers" ; but the "borders" section might be helpful
I ordered 25 plants from them/ they arrived with big healthy root systems and in perfect shape.

How about convolvulus? I love tricolor royal ensign because it is beautiful and easy to grow from seed. You do have to make sure you yank it out when dry because it gives you a TON of seeds. It hasn't given me any problems, great for full sun, stays compact and looks great as an edger. HTH's!