I'm facing the same situation...but also my first year with chickens, so trial and error.
but I would recommend against straw in a muddy run(from experience) once it gets trampled and packed down it will be like removing a very wet, very smelly mattress.
I have used shaveings from the coop on the bad spots in run, but not much better. It will absorb the wet quickly, and break down sooner then straw, but still a mess. It is much better to keep the water from getting there in the first place, then to try to dry it out. I also have a covered run, have dug drainage ditches around outside of run, and placed diversion ditches to get water away. Along with gutters on run, and coop.
But this spring with be major renovation of digging out all the "temperary" fixes in there, and packed gravel, with 6-8" of sand build back up. But most importantly ABOVE grade.
It looked like a pool in there earlier in the fall...
check out the "Muddy Run sticky" in the coop section--very well put together, and good luck.
TNT