Compacted, clay heavy soil! Partial sun! What to do? Cover crops?

Hello! I come to permies with a conundrum.

So my wife and I bought a house in the GA piedmont about 5.5 years ago and this is the year where we're getting pretty serious about gardening. We've got an acre total, but most of it is woods with the mixed pine/oak/sweetgum forests that so dominate the region. In the open parts of the yard I've had decent luck growing a handful of vegetables over the years with the addition of liberal amounts of compost I've had to till into the soil. But really, I'd like to just make the soil...suck less. It's super filled with clay and bits of sand, does not percolate well at all, and is just very compact. I presume pretty much anyone who's spent a reasonable amount of time in GA knows exactly what I'm talking about.

Obviously I don't quite have enough compost to improve the entire property...nor is it remotely practical/cost effective to just BUY a crapload of topsoil anyway (which would just erode anyway).

I can only assume that the logical solution is the obvious one: figure out what all I could plant to improve the soil. But honestly, I have no idea where to start. I suspect throwing down a TON of dandelion would help loosen things up and add a lot of nitrogen to the soil, but 1) I'm not a huge fan of the greens and 2) my wife isn't a huge fan of the way it looks.

So basically: I need suggestions. Are there any decent cover plants that would do a good job at repairing the soil, grow well in a part of the yard that only gets a few hours of direct sunlight a day, and also have "the look"?

Thanks!

I have the same soil as you do, worse...i get flooding at times.  I at least have some full sun.
I spent a few years with raised beds then wanted a real garden.  i tilled down as far as I could and got some big round hay bales and rolled out about 2 ft deep of tight packed hay over entire garden area.  The following spring i planted into it ( it was now only 5 in. deep). I planted a bunch of sweet potatoes, corn and okra through the heat of summer, all did okay. After gathering all crops I added another 2 foot of hay through the next winter.  The worms and rolly polly's as well as fungus turns that hay deeper every year into the clay soil. After 3rd year i could grow anything in there.Now i plant in early spring, plant buckwheat through the heat of summer, do a fall crop then even brassica's and greens through winter. this was very cheap and easy to do. If you start now you can grow some good crops next spring.

I too have a lovely clay yard.

I spread wood chips. I had a layer 4 inches deep over about 3/4 of an acre and it really helped improve the soil. Not enough for a garden but enough that you can walk around in the winter without growing "clay stilts"... I spread a bunch of seed (alfalfa, daikon, vetch, etc) over the places that actually are growing topsoil but it's too early to tell if I jumped the gun.

Really though, hay is probably better for a garden bed, like Ralph said, as it will break down faster. Wood chips will do the work similarly but will take longer. Just trying to throw out some more options. Wood chips are generally free and I like the look.

Dan, Ralph:

Thank you both for your feedback! After digging around, it turns out there's a waste management station about 4 miles from my house that'll give me all the mulch I could ever want.

YES!

I looked into the hay idea too, but free is free, and if it means that I've got to wait a while for it to break down...fine by me.