Groundcover, drought resistant & invasive

[Avalon]

Seeking suggestions for a groundcover for 7b (Georgia) for a few slopes, full sun, drought resistance. I want something invasive and fast growing. I was looking at Speedwell Georgia Blue (Veronica Umbrosa) and Blue Star Creeper and Red Creeping Thyme (Thymus Serpyllum Coccineus ). Would any of these work? I would like something under 5 inches.

[Marlingardener]

Consider dichondra (also known as Pony Foot) that comes in several varieties.
By "invasive" I guess you mean it spreads? Dichondra does that, and once it gains a foothold, you can't really get rid of it.

[laurie basler]

I love the Dichondra idea. No mowing.