Water should drain right through sand with no problem, so it has to be the soil base that you layed the sand on, or as suggested, it is the low spot in your yard, and water collects there. If it's the low spot, then perhaps the move is the only answer. Otherwise you need a drain system.
You could add a perimeter drain similar to what most homes have, but that will involve quite a bit of work. You would first have to dig a 1' wide trench all around the run, to approx 3 feet deep. (If needed a couple of cross trenches could also be added)
You should also dig a drain pit, just off to one side of the run. The pit should be dug deeper then the trench, say at least 4-5' deep, and be perhaps 3' X 8' in size. Dig another 1' trench from the drain pit to meet up with the perimeter trench, again about 3' deep. You always want your drain pit deeper then your perimeter trench.
Then lay your drain pipe (BIG O is the cheapest), in to your perimeter trench, (joining the pieces together as best possible) till you get to the trench leading to the drain pit. Use a T connector to join the ends of the perimeter drain pipe, then add another piece of pipe to the T to run down the trench to the drain pit.
NOTE. use a level, and as best you can, have the perimeter drain pipe at the far end, (furthest from the drain pit), slightly higher then the ends that lead to the drain pit. You can add a little soil under the pipe to accomplish this. The idea is that all the water that enters the perimeter drain pipe will drain "down to the pit". You want a slope to the pipe, even if it's just a few inches higher over the length of the run. When that is all done, then add drain rock to the trench enough to cover the drain pipe by about a foot. Then lay some landscape cloth strips over the drain rock, and finish filling the trench with some of the soil/sand that you removed when you dug it. Do this all the way to the drain pit. If your worried the chickens may dig down to the depth of the landscape cloth, I suppose bricks, or large rocks placed on it would stop them from ripping it all up.
The drain pit itself should have drain rock added to fill it to at least three feet, preferably more. Then lay landscape cloth over the drain rock in the pit, and backfill it with soil/sand, so it is level with your yard. (You can then plant grass, or whatever on it)
The use of the landscape cloth is to prevent silt/sand from filling up the spaces between the drain rock, which would make it useless eventually. It acts as a filter, allowing water to pass, but not the silt.
As I said it is a lot of work, and some expense. A small backhoe would be a bonus, or you could hire a neighborhood teen, who wants a few extra dollars, to do the digging. Once in place it should allow the water to drain away from the run, and prevent those big puddles.