After nearly killing myself "skating" across muddy pens on yet another wet fall day a few years ago & several chickens getting bumblefoot due to the mud, I knew I had to do something, even if it was just temporary to get through the winter. I have yet to come up with a permanent solution, but I have tried the sod idea the longest, so far.
I did sod my runs (have a few, too many Roos) but my runs are only 7x10 up to 10x10 each. I only have up to 4 to 7 birds per pen, too. More birds means more feet trampling, digging and scratching.
The sod worked great all winter into spring as we get a lot of rain here but I have very rich farm soil & prepped it well, churning it up 1st...yes, my back ached for 2 months afterwards.
When we get drought in July, the grass all dies, can't save it by watering, we get scorching 90s & 100 degree days Jul & Aug, so it's dry dirt pens the rest of summer. In the Fall Oct, our weather gets cooler. I have to scrape out the old layer of thatch roots from the dead sod, load into 5 gal buckets then haul to my compost area, then deep prong rake the compressed soil & add some lawn lime to prep for new sod. Talk about a workout! Rolling out sod is the easy part, prepping the soil beforehand is backbreaking & takes days. DO NOT BUY your sod Until AFTER the area is prepped & ready...important lesson here, trust me.
This is NOT a good way to do things, I admit that, but I have tried sand runs, mulch runs, straw runs & they were much worse with the vast amounts of precipitation we get here over fall, winter & most springs. Covering my runs to keep dry is not possible the way they're built already, besides a dry run is still just a dirt & poop run, still nasty.
I have too many predators here to free range. The only other solution would be a portable pen on wheels, known as a "chicken tractor". That way the chickens can be moved daily & the area recovers from the scratching & digging. There will be holes dug & unfortunately since the unit is mobile, a fox can dig right under. This won't work for me as I can't be right there 24/7 to watch for digging foxes. It might work IF you have a good farm dog dedicated to protecting your flock.
Anyway...it is a dilemma.
I've made wire panels to sort of lay down & cover the greenery section. I planted chickweed in a small section of pen, it grew up through the wire mesh cover, so chickens can walk around on top & enjoy eating what grows through, but can't turf up the roots. In theory, a movable cover "could" be used all over, but that isn't practical. Who has time & the back muscle to constantly be moving huge wire panels? I don't.
So...this is why I do the yearly "pain in the wazoo" sod. It takes a week of time & costs me about $200. Bumblefooted chickens & me busting my ass slipping in mud all fall, winter & spring is NOT an option, though.
To grow ANYTHING, you need rich soil for roots. Sand & clay are not going to grow anything well. Add hot, fresh manure from chicken fuzz butts on a daily basis...nope, can't grow diddly squat in that.
IF you decide to grow sod, or crop cover, you need good base soil 1st, & several inches of it, too. Your area is so large, you'll surely have an aching back, so get some help, prepping will be the most difficult part. You'll need truckloads of good soil.
Once you plant something, or roll out sod, usually have to keep all chickens off for abt 3-4 weeks & you must water well DAILY for roots to get established, if there's no rain. This is why I wait til Oct or early Nov when my weather changes here, cooler & very rainy so chickens would be in the coop mostly anyway then...and I have larger than average coops so they're not crowded or hot & hang cabbage & have pecking blocks, etc. When chickens see the new grass pen, they may want to turf it up, so that's why I have 2 covered dustbath areas...they go to those spots immediately to dustbathe after being cooped up 3-4 weeks. While they were cooped up I sprinkled them & the fresh wood chips in coop with ProZap to prevent any mite issues by the way...so yeah before I even start on the pens I clean the coop completely as they'll be cooped up in there a few weeks. I make sure no one has any mites.
By Oct here the days are much shorter, that means less time chickens are out digging & scratching, too. The sod does great all winter. The days gets longer in spring. They're outside more trampling the greenery. If it's a dry spring, the sod dies & may be all dead by Jun, but it definitely dies by Aug, just too darn hot here then.
I haven't come up with a better solution & this works where I live. When I do relocate, wherever I go, I will have to adjust to the weather. If I moved to AZ for example, sod would never work. Figure out your weather patterns for your area, sod won't work if you don't get enough rain. Know that WHATEVER you plant, will require some kind of maintenance or periodic adjustments due to weather.
One thing for sure.. chickens do need a dustbath area or section, preferably covered so it never turns into a mud hole.
They love scratching & digging, it's what they do. You have to watch for anything sharp that may injur feet or constantly muddy, poopy areas that can cause bumblefoot. Having various "things to do" or perches will keep them interested, like hanging fresh corn cob, cabbage, melons, a chickweed or pecking greens spot, etc.
Sorry I don't have a better solution. I don't know your climate either, so what works here may not work at all elsewhere.
The only other thing to suggest is dividing your huge pen into 4 sections, then you rotate birds & let the ground recover. That was actually my initial intention here but I started out with 4 chickens and had 3 pens to rotate them... Chicken math changed everything here, dramatically!

I have had up to 40 chickens, 11 of them were roosters... A few died of old age and I have not gotten any new ones because I have learned my lesson as to the number of chickens I can comfortably accommodate.
Best of luck!