The probably being going to move soon part is definitely a complication.
I'd suggest doing everything you can to prevent water from going into the pens and getting out what's there -- that means, gutters on any roofs including the coop, wiht downspouts directing water WELL away from the pen; and tarp the top and/or upwind side if you can do so without things collapsing or blowing away; and for sure dig a trench around the outside of the pen (not too close to the walls) preferably with an extension to lead the water away to somewhere downhill.
Ringing the pen with a trench really will help, especially if it 'goes somewhere' -- not only can it intercept surface flow, it will give water in the ground in the pen somewhere to drain to. All you need is a shovel
Unless the dirt you are digging up is horribly more clayey than what's already in the pen, you could even put the dug-up dirt in there to create a small raised area, although it will be extremely muddy for a while.
Second, see if you can get something affordable that will raise the chickens up out of the muck in the run -- coarsely chipped wood, maybe even pine straw (although it does have pointy poky bits). you won't be able to leave it in the run forever -- better rake it out before it starts to decompose and get really yucky, or if it starts to stench -- but it will get the chickens up out of the mud for at least a while, by which time *hopefully* your trench etc will have done some good.
Good luck,
Pat