That should help a lot. I don't tend to find mud soiling on my eggs even though we have a lot of rainy weather, as I keep adding litter to the run through the year and the birds tend to "wipe" their feet off going up the steps into the coop and through dry, clean coop bedding before getting in...
Depends on what is causing the dirty eggs. Rollaway nests aren't necessarily the wrong choice, but depending on the issue you may be able to resolve it without having to get new boxes.