Help with Frogfruit (Phyla Nodiflora)

Greetings.
I need help controlling unwanted plants in much of my front yard where I'm trying to grow Frogfruit (Phyla nodiflora Family: Verbenaceae) as a ground cover over roughly 1000 square feet. I am in southeast Texas (about 10 miles north of Beaumont), where the soil is a sandy loam (slightly acidic, which I've neutralized); this area of the yard is exposed fully to the sun. The rainfall here can be anywhere from too much (coming off the Gulf) to too little (and very hot). My interest is to plant a Texas native ground cover to replace most of the grass in the yard. This project is not well accepted by the neighbors, since most of the neighborhood is covered in St. Augustine grass, but it is especially unappreciated since I cannot keep it from looking "weedy" with all of these unwanted plants therein.

I started the project two years ago by applying Glyphosate to existing vegetation, waited for it to die, then applied fertilizer and compost, and tilled the area. The following Spring, I planted Frogfruit on 18-inch centers. The Frogfruit has been slow to come in, last winter it died all the way back to roots. This year the Frogfruit came in very well, as did two plant types that are quite aggressive and difficult to control as they invade the ground growing the Frogfruit. These aggressive "weeds" are (1) various forms of sedges, and (2) Virginia buttonweed. (Not to mention bermuda grass, but I've figured out a selective herbicide to control it.) Apparently a pre-emergent has little effect on this weeds. I have tried to pull (and pull and pull) these weeds, but now I'm reading that pulling these weeds can make them more difficult to control because these plants can grow from even a fraction of root or plant remains. So I tried a couple of selective herbicides, but that killed the Frogfruit as well. I've read that not watering can help to kill these plants, but I cannot control the rain.

I am at my wits end, and I'm seem to be out of viable options, and so I'm seriously considering abandoning this Frogfruit project completely because I cannot control these weeds. Any help would be welcomed.
Many thanks,
purpletalker

If you spray whatever kills the weeds, and use a shield, like a piece of cardboard, to keep the spray off the Phylla nodiflora that can help.

Any area where the two are growing mingled together is more difficult. If they can be untangled so the weed can be sprayed, do that.

You can also apply weedkiller via a paint brush to each leaf, but there is no way I could see doing that over 1000 sf of planted area.

Other option:
Start over.
Spray to kill the weeds, and do not try to save the Phylla. Do a thorough job on the weeds.
Water, (keep the weeds actively growing to circulate the weed killer)
Spray again about 1 week after the first spray.
Water. Give it a couple of weeks to see if anything will come back. Spray if needed.
Rototill, and hand remove every last remnant of plant in the area.
Water and give it another couple of weeks to see what comes up. Spray if needed.

This time, use weed mat.
Plant the Phylla through the smallest possible slit you can make in the weed mat.
Cover the weed mat with mulch.

Diana_K,
Thanks for the ideas...yes, the area and the entanglement is too much to try either spraying or brushing the individual weeds.

Question about the weed mat...the Phyla grows both stolons and rhizomes, how does the mat effect plant growth and propagation?

The weed mat I like is a soft plastic that is punched full of small holes. Not the pressed strands that are so stiff.
There is a lot of air and water exchange, so the roots under the weed mat will be happy.
As the plant gets established it will send out runners on top of the weed mat, through the mulch. The good plants will grow fine like this. True, they won't root so much through the mat.
Weeds are easier to pull, since they usually do not root through the mat, unless the seed falls just about on top of one of the slits you made for the Phylla.

What if you chose a ground-cover that formed a denser mat? Like one of the Stepables creepers http://www.stepables.com/scripts/prodlist-plants.asp?idLookUp=31
That would have the twin benefits of crowding out the weeds more effectively, and presenting a more even surface that looks more on-purpose and less "weedy" for the neighbors.

Here's the search where you can put in your exact requirements. http://www.stepables.com/3/find_the_right_plant_for_your_area.html

Phylla nodiflora is one of the denser ones, in my area. It will creep into lawn, and almost totally take over. But any mono-crop (growing one species only, and not allowing weeds) is going to be tricky.