I have gotten our stinky, sand based coop/pens to work w/ DLM. Both inside actual dirt floored coops , covered runs & in open runs. The trick, when getting a lot of rain & high humidity, is using different size particles/materials that allow poop & related wetness to sink in, breakdown but also has open areas for both water & air to percolate through. Also, with overload of rain, make sure excess water is deflected by french drains, berms & ditches - don't have to be huge, just enough to redirect flowing water.
Part of the issue was using only hay. It's generally the same size/length & you've discovered how it mats & becomes stinky/anaerobic. By itself, it is also not a very absorbent material. Use other materials, at this point, I wouldn't worry about mixing it in. Just bring it in & dump it. Leaves (it will take A LOT to get this started), wood chips might be best - all different sizes. Pine shavings - can help w/ scent. If dumped in piles next to each other, your birds will have a ball spreading & mixing it together. We've even used shredded paper - my shredder doesn't double cut, so it's larger, not extremely fine or small. Be aware, shredded paper can pack down when wet if only material & can/will turn into cement (think paper mache & papercrete). So it definitely needs other materials. I shred cardboard - different types/weight/sizes along with Bill's, junk mail & newspaper. I try to mix those up, too.
Here are pics over the years of DLM for us. We need many more bags right now again...
https://photos.app.goo.gl/3Z4ZpfpHr1Nr5DZC8
To see some pics of coops/runs - we don't have anything fancy, but doesn't stink. Pine straw & oak leaves are wonderful things... but there are many other materials as well.
https://photos.app.goo.gl/fgKYkhxNAXBnH6B36
That's interesting - from my phone, Galaxy 8 w/ Verizon service, I don't see signature blocks. We are in Cameron, NC - about midway between Fayetteville & Raleigh. Considered "the sandhills" region. This year, in 2019, we've had rain much more regularly but in much smaller increments (certainly not like in 2016 when we'd had 16" + of rain in the months of August & September - then Hurricane Matthew dumped a measured 11" on us in less than 12 hours).
https://photos.app.goo.gl/uY25ubH7hd3TzL3J8
https://photos.app.goo.gl/VmtUgwJww88W5t8L8
I didn't have good, deep DLM in that coop at the time - it had simply been way too wet to gather materials... didn't take long, after Mathew, to get those 3 coops going again (1 next to barn & 2 behind tree).