Down here in Georgia I have plenty of clay too. I have Mitchella repens and it is a great spreader. Like many plants, the first year is not much but by the second and third it was moving along. Love those white flowers and red berries.

I also have Galax and it is much slower to spread. In fact in most areas it is still just the original clumps. It is a beautiful plant and I love the bronze color in the fall, but I don't consider it to be a groundcover here. Certainly not invasive in the least.

Good luck!

Variegated lamium maculatum does great in my clay shade and is also drought and heat tolerant. The variety I have has yellow flowers are yellow for a few weeks in Spring, but there may be other varieties. It is evergreen in zone 7a. When it throws runners farther than I want them it's just a matter of pulling the ones I want to thin out or a little hand trimming.

I have two varieties of low sedum that are evergreen. Unfortunately their names have been lost. They grow in sun, shade and everything in between -- never invasive. You can explore it at local nurseries, both are succulent types. Good luck.

I live in zone 5 and need a perennial ground cover for an area under a clump of bamboo. This spot never gets sun, is always damp (or wet) and has heavy soil. I understand bishop's weed will grow there, but do not want it because of its invasive tendency.

Pachysandra is a terrific ground cover for shade. Not invasive, does well in wet or dry areas, evergreen even in my zone 4 garden.

I have had success with Pachy, a beacon silver lamium and periwinkle in my area.

I had lamium in Michigan and it was invasive. Is it not elsewhere?

Ajuga is nice. Some varieties are better tolerant of clay than others.

periwinkle, although pretty, is very invasive in my Oklahoma garden. I am constantly pulling it out and still it thrives.

I like periwinkle as well. It does spread, but is easy to pull out where I don't want it. I like the fact that it stays low and covers the ground, but does not bother the ferns and hostas that share its space. I also love all the beautiful blue flowers in spring that provide color while I am waiting for the hostas and ferns to emerge.

Here, pachysandra is the spawn of Satan.

See how sweet woodruff (gallium odorata), or one of the other woodruffs, behave in your area, I've found it to be well behaved here & it is shallow rooted so the underlying clay shouldn't bother it.