How have all the large animal keepers been keeping their pastures dry and free from mud?

Do you have enough land to rotate your horses? If so, that works really well (on about a 1-2 week rotation). The other thing I have done is rope off the pasture. It will make it smaller but If ind that horses tend to graze in one areay only. Make the pasture smaller to make them graze in areas they might not other wise graze in. Or...just rope off the wet sections until it dries. (you can just use the electric fence with posts)

Check with your state veterinary extension agent (free service to you by the way) and they can probably give you some tips, same for your state agriculture department. Drainage ditches can help, but they would know more about the area and the soil you are dealing with.