I'm a couple hours est of you. If you have the same soil as me, sugar sand, the soil test will show next to nothing for nutrients, little organic material, and a strong acid pH. Changing the nature of your soil will be a long slow process. The heat and humidity promote microbial activity. This will consume organic material in your soil at an astounding rate. Fertilizers leach through the sand with each rainfall. Compost can be raked in, but it breaks down rapidly, and is gone without a trace in a summer. Incredible drainage, combined with the sun directly overhead dry the soil instantly. For grasses, not many species will perform for the long term. Bahia, centipede, and St Augustine are the main choices. The oak trees shed their leaves in the spring, covering the soil and grass with slow to break down leaves having a pH of about 5.7 right at a time when the grass needs to be established coming out of winter dormancy. I feel your pain. I'm trying to grow vegetables in this stuff.
Nut Grass
Also known as nut sedge. Try sugar. Sprinkle it over your lawn, water it in. The sugar promotes those microbes to eat and breed. About a pound for every 250 sqft. Repeat every 3-4 weeks, works best in the spring. The microbes are able to reproduce in such numbers that they go after the root nodules, killing off the nut grass. Other grasses are not affected. This will leave bare spots of sand when the microbe population crashes, but so will herbicides. The sugar won't poison your kids for the next few years.
Oak Leaves
Gather these up. Grass has a hard time establishing in soil covered by leaves. They are a resource. Make
leaf mold and compost. When ready, these can be applied to your soil. In the meantime, the acidic soil could surely use some lime added. Adding lime will need to be done every year or two, but removing the oak leaves will help slow the acid creation.
Mulch
Rake it out from the shrubbery, tear out the nut grass, then mulch anew, good and deep. You can reuse the old mulch. Top it with more. Oak leaves make a fine mulch.
Mow
Gather those clippings, add them to the oak leaf compost.
Well water is much better than city water. At least you are not building up salts in your topsoil.