I realize I'm late to this discussion, but if that is what you were planning to do, there is more to do than that.
jp4LSU, where do you live? Are you in Louisiana?
Before you do anything involving gypsum or lime, be sure to get a soil test that will tell you which kind of lime, what amount, and when to apply. Grey clay might not have a single bit of clay. It could be a combination of other minerals and salts which mimic clay in density and water repellancy but lead you to use exactly the wrong chemistry repairs. Howard like the Texas Plant and Soil Lab for soil testing. They are good but I think you get a more comprehensive test for less money at Logan Labs in Ohio. Also I can get you a free reading of a Logan Labs soil test but not for a TPSL test. There is a guy named Andy who moderates a soil forum on another website. He and others there have been reading Logan Labs soil tests, thousands of them, for years. If you need exotic micronutrients to improve your salt balance or soil chemistry, they will tell you where to find it online, how much you need, when to apply, and more. Andy is all organic, so the chemicals he will suggest will only be in micro amounts to tune things up. And he will never suggest using chemical fertilizers.
Assuming you can loosen the soil, you will still need to add soil to the low spots where the water accumulates. If you don't improve the drainage there, the only grasses that will grow will be swamp grasses like nutsedge.
If you have not made compost tea yet, it is too late in the south. Water (air) temps have to be in the 60s or low 70s to get anything out of compost tea. What you can do is fill the leg of some women's stockings with compost and plunge it in and out of the coldest water you can put into a 5-gallon bucket. Plunge it in, squeeze the bag of compost, wring it out, squeeze the water out, and plunge it back in. Repeat for about 10 minutes. What you're doing is washing the beneficial microbes out of the compost and into the water. Then IMMEDIATELY spray that on the yard.
You will save a lot of money by using liquid molasses instead of dry molasses. They make 40 pounds of dry molasses by taking 30 pounds of waste ag material, like chipped corn cobs, and soaking it in 10 pounds of liquid molasses. 10 pounds of liquid molasses costs about $1.00 at a farm and ranch co-op. Find one in your area that sells bulk molasses. You bring in your own container and they fill it for the wholesale cost of molasses. It used to be $0.10 per pound a few years ago. A 1-gallon bottle will hold about 10 pounds costing about a dollar. You can spray liquid molasses with a hose end sprayer, a backpack, or an ag sprayer on a tow-behind. Spray at rate of 1 gallon per acre or 3 ounces per 1,000 square feet.
You can fortify the molasses by adding milk, liquid seaweed, and/or shampoo. Milk and seaweed add protein, minerals, vitamins, and enzymes. The shampoo will help carry the liquids including water deeper into the soil. The idea with the shampoo is to moisten the soil deeper down. That allows the beneficial fungi, the ones which normally soften your soil, to repopulate. Rates for all those materials is 1 gallon per acre (3 ounces per 1,000).
If you are going to aerate, take advantage of the holes in the ground by trying to fill them with water. Here again the shampoo will help the water to penetrate, but you need to get the soil moist down deep. If the soil ends up soggy for awhile, be sure to stay off of it until the moisture either soaks in or evaporates. Keep after it and the process will work for you.
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