Sand cures a wet run pretty well, provided you put down ENOUGH (typically 4-6+ inches) and put it on BONE DRY GROUND. If you dump sand onto wet ground, it will disappear into the mud within weeks or months and you will be back at square one with nothing to show for it.
Since your run is wet now, I wouldn't put snad in unless you can get it free and need the exercise. Best bet would be coarse organic matter like coarse bark shreds or coarse wood chippings, or pinestraw if you are not concerned about its pointiness. Put in a THICK layer. Basically you are mostly trying to raise the chickens up above the wet. Next spring you will need to rake out all the organic stuff, before it starts to decompose and make the mud worse!, and let the ground dry out real well before adding a good layer of sand and/or gravel.
Also, do what you can to limit water input. Gutters on the coop or other adjacent buildings, with downspouts directed well away (downhill) from the run, and a trench dug around the outside of the coop and run to collect water and divert it downhill, will often go a long way towards helping.
Good luck,
Pat