Floridian here- I think you are struggling due to the clay soil preventing water drainage. When I previously lived in a swampy area of Florida, everyone invested in truckloads and truckloads of sand/fill. And ponds. And water pumps to remove the water from the property.
You may want to consider digging a pond to capture the water, removing the mud you currently have and replacing it with something that drains. My neighbors invested in so much sand for their property, that they effectively drained their property into my backyard (prior to us moving out).
I have 4 horses on 3 acres currently and since we are on a hill, everything drains away to the sinkhole in my neighborās yard. No matter how much rain we currently get, the property stays dry because everything is on an incline. And sandy soil definitely helps.
Of course, then you have to worry about sand colic. I put mats around my feeding areas, but keeping those mats clean is work in itself.
I think I would move the horses to a different paddock. Some sections of pasture just seem to retain more mud than others. If moving them begins to create too much mud in a different pasture, then move them back to where they are currently. But it definitely sounds like the pasture they are in needs a break.
Good luck!