Not a short term answer, but things to consider down the line:
- Adding bricks, pavers, rocks to the run gives them the option of whittling their beaks and nails (to some extent)
- You need to look at what can be done to address the mud issue. I use deep litter in the run because I previously had a major mud issues. It's unavoidable that we're soggy all winter but the chickens have reasonably clean feet as I add a lot of extra litter during fall.
Short term you may need to trim nails if they're truly overgrown. Generally dirty feet aren't an issue but I don't know how bad it is.