I've used the deep litter method in hot subtropical conditions which coped just fine with serious monsoonal weather, complete with massive flooding and rains which persisted for months.
Some people have tried and failed with the deep litter method and now advise everyone against it, but it's worked really great for me and since I moved house a few times, I've restarted it everywhere they've been and find cleaning cages is no substitute for deep litter.
I actually started the deep litter method because, well, a few reasons: every time I cleaned out their coop they got sick, same day, every time; these were old coops which had been overcrowded and which the previous owners had let animals die and rot inside of, the floors were diseased. It became obvious that cleaning out the floors was only allowing harmful bacteria etc to flourish, since that's what recovers fastest, whether you're just cleaning or using harmful chemicals, or both.
So I cleaned out the top layer one last time, limed, (may have dug it in a bit and watered it in too, don't recall offhand for sure bit think I did), put down hay, and basically repeated the re-application of lime and hay for a few months afterwards, and there was never again a whole-flock malaise as there had been every time I cleaned the coops --- plus, it got stinky real quick after cleaning, every time, whereas the longer I left it without cleaning the less it stank. Soon it became clear that liming and deep littering was the only option for these coops.
Also, I lived right next to a subtropical forest at that point which supported a massive overpopulation of animals perpetually, including a feral dog pack numbering at least 50 animals, a feral pig population also numbering at least 50, a fruit bat population of two species numbering up to 250 000 at times but with a base level of 10 000 at any given time, and many more animals besides including domestic ones and natives; that was a 25 acre forest and only the innermost areas were heavily used... Point being, if that forest could cope with such a massive permanent residence of so many species, and not stink, surely a little chook coop could to. The same applies to forest floors everywhere, there are natural controls for fecal matter etc. Anyway, I used a garbage bag full of composted litter from that forest in the coops and it went great.
I'd probably just get the floor built up so it's higher than the surrounding areas if it's really swampy where you are. Perhaps a clay or natural (untreated) wood base would help keep excess water under it, and you could build more drainable layers on top of that using (for example) sand, forest mulch, hay, dirt, whatever.
It will likely take about 6 months before you will have a truly strong deep litter composting system going.
The first few months I added more lime every time it smelled a bit, but for years after that there was no odor except a mild and healthy soil scent, nothing like ammonia, feces, molds, etc. Years later I dug experimentally down into what had once been rock hard, unhealthy stinky soil and it was soft, rich, slightly damp and healthy and they didn't get sick again even when they excavated as well; on this note I don't believe 'dessicated' is the sign of full health some people think it is, since truly healthy soils are rarely completely dessicated, and jungles never so. Chickens are naturally a jungle species after all, it's not like you're taking a desert species of bird and forcing them to live in damp conditions. My turkeys had issues with the diseased house paddock but not the cages, and once I learned how to treat their issues they were fine. The cages were never an issue for them. The top layer was almost always dry but when it rained a lot and got wet, even muddy sometimes, it still didn't stink. When I brought in new chooks I'd lime the soils again to help control shed pathogens, oocysts etc but that's all it needed after getting started to remain healthy.
Lime is indispensable to this in my experience. You could probably do it without lime but it'd take more know-how. (When I say lime I mean just plain calcium carbonate, not one of those ones which burns the chooks... The actual ingredient is important to know, as various limes are sold under the same name in some places; 'AgLime' here is plain calcium carbonate, for both soil treatment and animal feed addition, but in the very next state it's caustic and unsuitable for either but still marketed under the same name. Best to check the labels carefully, make sure it's plain calcium carbonate only).
It helps kill the odor causing pathogens, bacteria, fungi etc and allows their beneficial counterparts in the soil to take over, helping the soil become healthier and alkalized, and if you add in an inoculation of forest mulch that brings in all the healthy microbes and so forth which can populate the litter and keep the potentially pathogenic population under control.
If the flooring has been sprayed with some seriously harmful stuff it can take a long time or even be impossible to get the litter healthy since the soil or flooring is toxified and only the most harmful organisms etc thrive under such conditions. If it's concreted over, that's another issue, drainage is pretty important to soil health overall, but since you're in a swampy area you likely already have all the right soil constituents to thrive under such conditions.
I'd also advise against blindly following any formula 100% as your conditions are unique to you -- everyone's conditions are unique in some way. Better to play it by ear and experiment if it's not working for you after about 6 months; learning about soil health is incredibly helpful as well and is one of the best ways to guarantee your plants' and animals' health (as well as your own as the apex of the foodchain, obviously).
Some people who say they use the deep litter method are in fact using something akin to it, often on dead/poisoned soil or concrete with zero drainage, and cleaning it out every few months yet still calling it the deep litter method. It's not quite the same thing. Deep litter composting is generally what is meant by the deep litter method, not litter buildup or slow removal practices. I also wouldn't use pine shavings etc as the volatile oils in pine are pretty caustic and harmful to many soils which have no native microbes etc to handle such oils or chemical compounds.
Best wishes.
