Hi - Can try the ground cover
vinca vine or ajuga / bugleweed they
both flower and are evergreen.
I am in z-8 Texas. Mine do fine here.
Do try to remove the grass.
Good luck. tina_2

The toughest groundcover for dry shade is Geranium macrorhizum, Big Root Geranium. It's semi evergreen, needs no attention, is coming into bloom in zone 6 now. It establishes easily, is also deer proof, and very attractive. Set rhizomes every 12-15 inches for fastest cover.
If the grass is dead, you won't have to remove it.

I agree, you cannot go wrong with Geranium macrorhizum.

It is 3 season, one of the earliest things to be green in the garden, because it IS semi-evergreen. Has a nice, subtle pine-like scent. It is SO easy to plant, just cut in a bit with a spade and put it in. No major digging. For the most part, normal rain will be enough water to get it started. If you want it to start even quicker, give it some extra water.

Has pretty fall color. Blossoms colors include whitish pink. That's probably the most commonly found one, called Biokovo. Personally, I prefer my pink one, called Ingwersen's Variety. But, if you are in zone 5, there's another pink one that you could grow, which I cannot: Karmina. Look at bluestoneperennials dot com, they have it.

Here is a link that might be useful: ingwersen's

I think that 'Biokovo', 'karmina' and 'Ingwersen's variety are hybrids. The straight species, G. macrorhizum, is a larger more vigorous plant with fuschia colored flowers. There is a variegated form, and I think a white variety that is very hard to find. But the cultivars you name are distinctly different plants. And not quite as deer proof.