We recently filled in the base of our open-air coop with deep sand. We have clay soil - i.e. very poor drainage. Over the last few years, the coop floor fell below the surrounding ground level, which made the situation even worse.
Now the coop floor is sand and several inches higher than the surrounding ground level. It is so much better - dries quickly after the rain comes in from the side! I periodically scrape off the surface poop, which is dry and easy to collect. I am still looking for a better tool than a manure shovel - need a giant litter box scoop! And of course, I should do the cleaning more often than I do. Nonetheless - the sand floor is a vast improvement on our old dirt floor or shavings in our environment.
I originally got the idea from the Greenfire Farms website. They also are in Florida and have open air coops, so rain comes in from the sides. You can see their coop construction on their website - look at the blog titles. Since they have such pricey birds, I figure that they wouldn't use the sand floor if it created health issues, similar to sand colic in horses. So far, I am not aware of any problems in my birds related to ingesting sand.