I'd go for the 'deep litter' rather than more sand.Well, spread a bag of PDZ in the run and it seemed to help. My husband with the sensitive sniffer seems to think it doesn't smell as bad as before. Tossing around the idea of raising the run and bringing in a deeper base of sand come warmer weather. May do a side by side experiment with deep litter vs. sand. I'll keep updating.
I used sand as a brooder bedding, which worked great for daily cleanups, but eventually the sand become saturated with pulverized feces and you can't separate them or clean it. The sand was good for nothing but filling some holes in the front yard from an auto accident.
Sand can help with drainage but it can also hold moisture, as you've found.
When it comes to chicken poop, moist equals stink.
The DL can also hold water but will also support a microbial population to 'eat' the feces that sand can't.
If you do decide to try the 'side by side' comparison, better put a tall(~12") barrier between the 2 as the chooks will fling one side onto the other.....
......or you can evaluate the 3rd option of a mix between the two materials.
Best of cLuck to ya, looking forward to hearing about how it works out for you.