We knew the entire area where I wanted to build my coop was low. I had clean, fine sand delivered first, and we put it where we were going to build the coop. Not in the surrounding area. Yes, the sand was expensive. We made it at least 1 1/2 foot deep with the sand. I then had clean fill dirt delivered, and put it on top of the sand. Once again, at least 1 1/2 foot deep. Now that the rains are coming in, and that area is swampy, my coop is high, and dry.
There are 2 reasons I put the sand down first. 1. I can hose down my coop, and runs occasionally, and sand is a good filter medium. No odor, and no contamination. 2. When you elevate a low area, you don't really displace the water. The water is still there, just beneath the surface. Fill dirt alone, will wick, and eventually become mushy. The sand doesn't wick the water, and it won't become mushy.
Before we bought this place, I was renting an area from a person, for my coop. His whole property was very low. For the most part of the summer, he could not use his driveway. I had to park by the road, and load a wagon with my supplies, and haul it to the coop. I offered to help him with the driveway, and have a few loads of sand, or pea gravel delivered. Through a friend, he was able to get about 7 dump truck loads of fill dirt, for free. He elevated the driveway quite a bit. I told him I didn't think it was going to work the way he thought. I told him that if had he brought in enough dirt to elevate the entire low area, it would work, but just elevating the driveway, he needed sand, or pea gravel underneath the dirt.
Yes, when the rains first came, he was so proud of himself, his driveway, and I was wrong. After 3 weeks of rain, he couldn't use his driveway. The fill dirt got mushy, and some of it eroded.
While you don't have to have a totally dry coop/run, at least consider using sand, then fill dirt in a portion of it, so the chickens have a dry area. Fans in the coop help with heat, and mosquitoes don't like the breeze it creates, so they help some with mosquito control too.