You might try reading this.  
Pats Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
Do what you have to in the short term.  Maybe putting something out there to act as dry islands above the muck may help.  Any long term fix probably requires dry weather.  You may not see that for a while in Oregon.  
There are a couple of different general principles to follow.  Try to keep the water from getting in there to start with.  Use drainage ditches or berms to keep rain runoff from going in there.  Make sure your coop roof does not slope so rain runs in there.  If you can cover the run, fine.  The problem with a big run is that they are hard to cover.  
I used to camp a lot.  Putting up a rope and draping a tarp over that works real well to help keep rain off.  A possible problem longer term is that high winds can cause damage, but do what you have to.  There is a trick to tying it off so the wind does not pull out the grommets.  Take a small wood chip or gravel and fold and twist it into the corner, then tie around that.  It is a bit hard to explain, but that is a lot stronger than tying to a grommet.
The longer term fix is to set it up so it will drain.  Dumping a lot of sand in there will work for a while.  Sand drains real well, but it will wash away or work its way down into the mud.  You might need to put something around the fence a few inches high to keep the sand from washing away.  Putting a layer of gravel down under the sand will help keep it from getting worked down into the mud, but I suggest trying to use rounded gravel, like pea gravel or the gravel you get from a stream bed if you can.  Crushed gravel with sharp edges can cut their feet when they are scratching and cause bumblefoot.  It does not happen that often, but it happens often enough that if you can economically use rounded gravel, you are better off.  But you need to do what you need to do.  
Good luck.  It is not an easy problem to solve with a decent sized run.  Mine still gets muddy in many spots when it sets in wet.