One thing I have learned many times over in construction is that if the weather and mud aren't cooperating, you just have to wait it out. You can plan stuff to stay ahead of the weather, but once the rain/snow/mud does it's thing, you just have to wait it out.
Even if traction wasn't an issue, you'd find that sticky packed fill will make a mess in your front bucket and box blade and neither will work they way they should (i.e., won't scoop or dump cleanly in front and won't spread in back). There are even times when damp fill will defeat a box blade, in which case I spread what I can and wait for a few hours of sunshine to dry it out some.
For foot traffic, I have found that putting down cardboard or plywood scraps will help. But a tractor or other heavy equipment will make a mess of that in no time.