The Deep Litter Method basically means turning your run into a compost pile. The microbes that eat the organic material (poop and bedding material) need some moisture to live and reproduce. If it is too dry they die of thirst. If it is too wet the anaerobic microbes take over. The anaerobic microbes stink to high heaven and are slimy. What you are aiming for is the Goldilocks zone of damp where the aerobic microbes can thrive.
Sand is permeable, it has great drainage characteristics. But for it to work the water has to have someplace to drain to. It sounds like your run may be on an impermeable clay that holds water. It may be in a low spot where water drains to it or maybe you dug a hole and filled it with sand. The water still does not have a place to go. One possible solution is to put in a French Drain or something similar to drain that water to a low spot, but that depends on the lay if your land and where the low spots are.
You can try putting a mulch on top and turning it into a compost pile. Some people have had great success with that. @aart has had great success with wood chips but I don't know the details of how that was used or the initial conditions. Others have tried using an organic material and it did not work, it still went anaerobic and needed to be removed. Sometimes it works, sometimes it doesn't.
Good luck!