The Whole Place is a Mud Bath ARRRGHHHHHHH!

Well rotted forest mulch (tree shreddings) as advocated by Foxy is the only solution in these conditions.  Confining the chx to surfaced areas and resting your late spring to early Autumn runs if you have the room.   Useful things are duckboards made from pallets cut in half so the chx can walk on these between the house and any field shelters.  Place their drinkers on these.  A thick layer of the mulch round the pophole with slatted duck boards on top and a covered entrance tunnel run to the house to keep it substancially dry will stop the mud being trailed in the house.  Create walkways for you from either mulch with duickboards on top or those border edgings which are made from half round logs connected together with wire make good walkways.  Don't image turf will survive so keep them off as much as possible.  Don't use straw other than as a tempory measure it rots quickly and can become foul smelling and slimy if left too long.

Foxy's system sounds excellent and well thoughtout.  It never ceases to amaze me when I read on this site about new chicken keepers who think they can keep a dozen chx on a tiny piece of garden with no spare runs.  All seems well during the summer but you have no idea how sustainable it is until you've survived a wet winter.  My winter runs are on the side of a hill so I get run off the slope  yet they are still trashed by my chx.  Fortunately my spring & summer runs recover well but it's a battle.
HF

As others have said, as you've such a large area, best to divide it, and leave one half to grow, while they destroy the other half.  Then switch over and reseed.  Lots of rain is great for grass, as long as you have reasonable drainage, should be fine.
Spring and Autumn best time for germination, so you'll have to wait a bit longer I'm afraid. In the meantime, you can be constructing the fencing and perhpas include a few field shelters, so if they want to keep dry they can.

And remember, people pay a lot of money for the benefits of a mud bath . . . .   So not such a bad thing really!