Creative lawn solutions

One of my favorite places to be is my back yard. It has a giant walnut tree.  It's the biggest one I have ever seen.  The tree gifts us with wonderful shade.  I have several gardens. Veggie garden, hugelkultur, a beginning of a food forest, a rose garden, several flower gardens, and several fruit and nut trees.  What I don't have is a decent lawn.  We have been content to mow weeds and call it good.  This year under the amazing walnut tree most everything has died.  Now the foxtails are showing up in force.  I don't water our pretend lawn. Being a California I use the water for the things I want to grow.  I hate!!! Foxtail.  I was reading in another post the way to get rid of them is the crowd them out with grass or other dense plant of choice.  My problem is it is already getting pretty hot. It would be extremely hard to plant grass this time of year. Not to mention a great deal of water.  I have several areas with wood chips, and my son said we should do that ( which is a total victory for me because I have been battling the whole family with my wood chips. They are starting to see the benefits. Mom isn't totally crazy after all). I don't really want that for this area. We use the fire pit here. It's a place we gather.  
At this point we are mowing it to try to cut it before it forms the nasty heads.  This is a total pain because they seem to be producing the head shorter and shorter. Also mowings this area is like creating my own personal dust storm.
I have tried planting short chamomile, but it died.  The area gets little sun from around now through about December.  I water to keep the tree alive, but water is scary low already.
I don't expect miracles, but I'm out of ideas, and wondered if anyone has a creative solution.  For now, or even for the future.  Thanks.

I'm not sure about the particulars of that plant, but is sheet mulching an option? Like cardboard, or even old blankets?

Hi Jen.
That's foxtail grass is it you have?  Sounds like it maybe time to get rid of the lawn!
I don't know enough about your climate to make a really good suggestion, but found a list of plants on laid back gardener, that may have something that would work as an alternative for you.  I would also suggest sheet mulching for a complete season, then spot plant some suitable low growing, preferably edible, creeping plants instead.

I'm N. California zone 9 b. Hot many summer days over 100.  We are dealing with 93 -97 this week.  Very dry.

I would consider Buffalograss, Buchloe dactyloides

It is drought-resistant and loves heat.  I have read that it is also shade tolerant.

I would try it at the drip line first then work towards the tree to see if it will tolerate walnut.

Another benefit is that it does not need mowing in my climate so it will not need mowing in yours either.  It is a perfect permie solution.

Thank you so much Ann. I have never heard of Buffalo Grass.  It sounds like the perfect solution.
I may cover the ground with cardboard, and wood chips to get it ready for the buffalo grass.  I'm not sure yet.  But I definitely want to give it a try.