If you aren't dealing with freezing conditions I've never had a problem with rain. Except it is hard to make a good looking set in the mud. Dig you some dry dirt or at least semi dry now, to use when you make your sets, the rain will wet it and blend it all in. I do understand your guys clay bakes like concrete and is basically the same as us northern guys dealing with freeze, but as long as that doesn't happen (sounds like you don't have to worry about that) and it isn't freezing, the traps should work just as well in the wet dirt/mud. Obviously don't set them where the water is going to pool, but I use wax paper for pan covers and it drapes down around the pan like Warrior said, and cover well with dirt, make sure you don't have a bunch of voids under the pan cover between the jaws and the pan for the covering to sink into when it gets wet and I've not had problem with several inches of rain. Of course dirt holes are bad to fill up with water, but flat sets should be good, try and put the lure on the underside of the backing so it is somewhat protected. Flat sets are often not in bare dirt anyways, so you will blend with a little grass, pine needles or whatever is there over the dirt and this will help if you have some minor sinkage of the dirt also.
I love trapping with native dirt, no need to pack waxed sand or anything to the set. Unfortunately 90% of my dirt trapping is in freezing conditions so I don't get to do it that much.