Something you could consider is Mondo Grass. It's a grass-looking ground-cover, that generally is evergreen (it got knocked down a little over the winter, but I cut it back and now you can't even tell). It can take a good amount tread. Of course, mondo grass is ornamental, so it isn't cheap like a bag of grass seed.

Something else you could do, with the mondo, is get the stepping pavers/slabs (from like Lowes or Homedepot) and space them out in a grid. Let the mondo fill in the space in between them. Just search "mondo grass" on Google, and you should see, in the images, what I'm talking about. I prefer the ones with the shorter grass, verses the taller ones... Walking through the taller mondo is like walking through grass that needs to be mowed.

I have the same problem in our area "over the creek". It's horrible hard clay, and stays soggy a long time when it rains a lot. I'm hoping someone can give you an answer.

I planted some ajuga in a bed several years ago and it has spread into the lawn. It's in the sun, though-don't know how it would do in the shade. Some of it has been under some bushes and looks good-it's dense so it has a good carpet effect.