Checkerberry (Gaultheria procumbens) might work for you. Also known as Wintergreen. It spreads by underground stems, and has dark green, evergreen leaves, and white flowers in summer. It also has the added plus of little red berries in the fall. It tolerates shade but grows and flowers the best with a little sun, but not all day hot baking sun. Morning sun and afternoon shade would probably be fine. The leaves take on a slightly reddish/purple-y color when the weather gets colder.

Checkerberry needs an acidic soil, so you may want to test your soil PH with one of those little test tube kits from the garden store. It likes soil that's fairly rich, like the kind you find in forests.

The only drawback to Checkerberry is if you have deer or other critters in your area, they may munch on the leaves and berries in the winter.

It also spreads fairly slowly as compared to Japanese pachysandra or Vinca minor. It'll take a few years for it to really start looking lush.

Wouldn't try to grow it if you're west of the Rockies, since it's native to the Eastern U.S.A. You can usually find it at a nursery that specializes in U.S. native plants and/or order it over the net.

Vinca became very invasive in our yards, and it became a landscaping nightmare for us. It requires friends to bring their Bobcats and digging it up is a major chore. We have to have it removed totally before it kills all of our trees.

my favorite ground cover is bigroot geranium, I grow it in your conditions, does great.

It's called Geranium Macrorrhizum.

I have the pink, Ingwersen's Variety. The most commonly found one is white with pink overtones, called Biokovo.

NOT Invasive, and has such a shallow root (literally on top of soil, you can pull it right off and give it to a friend.)

I have a patch of Vinca Minor variegated growing under an evergreen tree, and it hasn't been at all invasive. Very easy to control, maybe once a year I trim it back with a weed whacker. The only thing invading the lawn are the tiny patches of violets everywhere.