Builder then carefully planted grass, covered with straw, and staked fabric. This grew well until summer got too hot, held almost all winter, then shortly after new year the whole thing went again.

This spring, we are going to terrace in 3 - 4 steps. Each will be appx. 2' high and 6' - 8' deep. The barrier, for economical reasons, is artificial. It looks ok, but not stone or brick like rest of landscaping. I would like to find a plant that vines and produces roots like nobody's business in order to fill the terraces and drape over barrier.

In my opionion, it can't be too nasty or invasive. As long as it doesn't stink or isn't poisonous. :)

There are other portions of the slope that are out of play here, but I would still like to plant. These areas are rocky with a lot of slate and sandstone. While these areas have lost a few yards of sand over last two winters, they are hard enough to hold shape for many years. Still, anything green will look nice.

Any suggestions on such plants would be great.

When searching, you may want to look for aggressive plants. My preference is for natives. If you do choose an "alien" please make sure it doesn't fall in the invasive category.
invasive plants aren't eco-friendly. The problems that these nasties cause will reach much farther than your land.

Invasive Species:

"An alien species whose introduction does or is likely to cause economic or environmental harm or harm to human health."

good luck

If you want a groundcover that is very aggressive, St. John's Wort, hypericum calycinum will do it.

Another groundcover shrubby plant is cotoneaster. Some states consider it invasive (birds eat the berries and pass them along, so to speak).

Creeping thyme is another good one for full sun.

Creeping phlox (phlox subulata) is also great and colorful.

Clematis vines, though not evergreen, provide nice blooms.

If jasmine works in your area, that's another possibility. The foliage is evergreen here in NC.

You can tuck sedums into cracks and crevices of walls with little pockets of dirt.

Again, zone permitting, delosperma cooperii (ice plant) is another full sun possiblility. Evergreen here, loaded with sparkling flowers all summer (they close at night).