Can you post a picture of your crabgrass? Because all the crab grass died last fall and should have disappeared by now. You might have a different weed, so to help you with that we'd need to see it. Best pictures are taken on a cloudy day or in the shade. Best identification pictures would have a seed head on the grass plant. Why do you want to aerate? Is your soil hard? When it rains is it still hard? You can use starter fertilizer, but I would wait until you have roots from the new grass before fertilizing. When you fertilize in the early spring, you will have to be out mowing several times per week to keep it under control. Spring grass usually grows very fast without fertilizer, so when you fertilize, with chemicals, it just goes that much faster. So I'd wait until you are mowing the new grass at a fairly high setting. That might be late April or even late May the way this spring is going.

Mulch mow the grass at a high setting. This encourages deep roots and also helps to crowd out weeds. Usually, I'd say to water deeply and infrequently, but you may not be able to do that with your water situation. It would be better to water a section of the lawn for a long time, then another section for a long time on a different day than to water both sections a short time on the same day. If possible, it's best to water about an inch a week (adjusting for any rain that falls) applying it all in one day. In your case, you'd want to give each section an inch, move to a different section for the next inch and so on. In the hottest part of the summer, watering twice a week for 1/2 inch may work better. Fertilize once in the spring and twice in the fall, with the last one after the top growth has stopped but while the grass is still green. Those are the most important things to do. If you want to try to start killing the weeds, either get a weed hound and pull them or spot spray with something that has 2,4D in it. Don't use the weed and feed products. Fertilizing and killing weeds are different activities and should be separated.

The brown spot is probably from overwatering. Cut it back and the situation may resolve itself. Light, frequent watering encourages very thin, shallow root systems that can't tolerate a drought. More or less, it doesn't grow roots because it doesn't have to. During hot periods...most of the time where you are, I think!...twice a week is very much enough, 0.5" to 0.75" at a time. Cooler periods are around an inch all at once, once a week. Subtract any rainfall from that total. Clay soils do have trouble absorbing water. Mine does. Fortunately, for a lawn that small you can use one single sprinkler and cycle it, 15 minutes on and 15 off, to give the soil time to absorb. On the up side, clay soil holds a LOT of water once it does get wet. The time you water doesn't matter so much, but with brown spot please avoid watering in the evenings. That just encourages it! For now, stop watering. Once the grass shows signs of stress (when you walk, it doesn't spring back up in a few minutes), water it as above. You'll have to test to see what you're putting out--a tuna can works well, or an old coffee can. You'll slowly adjust it to longer waterings, deepen the root systems, and waste a lot less water. Quite a bit of the 5-7 minute watering is evaporating off and getting lost. For seeding, the rules differ--2 to 4 times a day watering, about 10-15 minutes at a time to get the seed going. Fertilizer...a soil test would tell you what you really need, but barring that a good balanced fertilizer in the 4-1-2 proportion is a good bet. 16-4-8 would be perfect, but if you can only find, say, 24-2-6 then use that. Just get as close as you can. Fescues are already some of the most shade-tolerant grasses, although others will have much better suggestions than I would. I'm a Kentucky bluegrass person myself.

A soil test is going to tell you that you have sand with no nutrients. Let's get some grass growing first. St Augustine loves a little shade. Bermuda, the other popular grass in FL, will not grow in any shade. For full sun the best one to get is the Floratam variety. There are a couple tricks to growing St Augustine. First is to mow it at the mower's highest setting. Let it get up to 5 or 6 inches tall before mowing it back down to 4. Second, it absolutely must have deep water every so often. When temps are 90 or above, you will have to water a full inch every 5-7 days in full sun. This time of year you might get away with watering every 3-4 weeks (always a full inch as measured by cat food or tuna cans). Forget about worms. They might come later or might not. The best thing to improve your sand is to use organic fertilizer. If you really have fine, white, sugar sand, then any amount of compost you can afford, up to 1 cubic yard per 1,000 square feet, would be helpful. My favorite organic fertilizer is alfalfa pellets (rabbit chow). Apply at 20 pounds per 1,000 square feet as often as you want to or can afford. Grain type organic fertilizers take 3 full weeks to see any improvement. They are called "slow release" but that's sort of a misnomer. They release fast but not to the plants. The fertilizer feeds the soil microbes and they feed the plants. Being a biological process it takes time. Over time your white sand will appear almost black. Don't be fooled into thinking your sand is changing. What's happening is the microbes and grass are leaving behind organic matter that will help to hold moisture and provide other benefits to your soil. As for the snails, find some cheap ammonium sulfate fertilizer and scatter that around. Ammonium sulfate looks like rock salt and has exactly the same effect on snails and slugs. They bubble up and die. We had a big infestation back in the 90s. I used that stuff (before I became organic) and we haven't seen a snail since. Incredible!!