I applaud your desire to use native plants. It is better for the environment, for native wildlife including birds and butterflies, and will be more robust and hardy for you. It does make plant selection a bit more challenging.
You didn't say if you intend to be walking on this ground cover like you would a lawn?
For a shady to part shade area, a very nice evergreen native ground cover is partridgeberry (Mitchella repens) . It stays about two inches high, gets pretty little fragrant white flowers followed by red berries which birds like (hence the name). It requires acidic soil and is not very drought tolerant and also will tolerate only minimal foot traffic.
Virginia creeper is a very adaptable vine which can also be used as a ground cover. It gets beautiful fall colored foliage, grows fast, tolerates sun/shade, gets berries that birds like (but are toxic to people and pets [but they are painful to eat so pets usually don't eat enough to be harmful] ). You don't want to let it grow up buildings because it suckers and can be damaging.
If your area is going to get a lot of foot traffic, then you probably need to think about native grasses. These would include Junegrass, little bluestem, grama grass, switchgrass. These will not give you the smooth golf course lawn effect, they grow clumpier than that. But they are tough and hardy and don't have to be mowed (or you can mow every once in awhile to keep it evened out a bit. This is little bluestem, left unmowed:
Other possibilities that aren't quite your typical ground cover include bearberry (kinnikinnik) which is a is a low growing, spreading shrub, that only gets about a foot tall. It is evergreen. It is a little slow to establish in the first year or so, but then will spread well and be exceptionally hardy and low maintenance. Juniper is also native for you and has prostrate, spreading forms.
You could order any of these on line or look for a native plant nursery near you.