Since the soil that grass is trying to grow in is the most important part of the whole equation take a good look at what you have and determine what needs to be done to make it into something good and healthy that will grow a strong and healthy lawn. These simple tests can help,
1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. For example, a good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains� too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.

How to keep weed growth under control will depend a bit on the type of grass you will be growing, St. Augustine, Bermuda, or something else, but Corn Gluten Meal can help with newly germinating "weeds".

I have not lived in WPB but I visited once. If I am correct you are on pure white sand. And I suspect you are growing St Augustine turf. With that in mind, here's what you do:

NEVER let it dry out. St Aug dies rather than going dormant from drought. Sand is touchy but certainly not impossible, especially near the water. This time of year you should be watering ever month unless you have received an inch of rain. Water all at once and water long enough that you can go for some time before watering again. The very worst thing you can do is water daily. Daily watering promotes weeds.

Mulch mow at your mower's HIGHEST setting. St Aug NEVER needs to be mowed at a lower setting. Low mowing promotes weeds.

Fertilize on the federal holidays. Because you live in WPB, you have the opportunity to fertilize on New Year's Day. Then Washington's Birthday, Memorial Day, 4th of July, Labor Day, Thanksgiving, and NYD. Read the Organic Lawn Care FAQ if you have not already.

Corn meal is not for weed control. Corn GLUTEN meal is for weed control. Those are very different materials from the corn plant.

If you do the above, then your grassy weeds should disappear. However, St Aug is notoriously susceptible to broadleaf weeds like clover, dichondra, spurge, oxalis, dollar weed and others. Some of those are easy to pull. Dichondra is just about impossible to pull because every leaf is its own plant. Good news is dichondra usually will die every summer from insects. If you are following the watering and mowing regimen above, and you still have broadleaf weed problems, then you might want to use a spray to spot spray the weeds. Those sprays go in the face of your wife's wishes but they are minimalist. An alternative to commercial weed sprays is vinegar. Unfortunately the over spray from vinegar will also kill the grass. You might try mixing some foaming hand soap in the vinegar so it comes out foamy and doesn't spray all over everywhere.

You might consider getting a professional soil sample done. Yes you have sand, you already know that. But you can go a ways to improve the sand and set it up so the grass will do well for you. In the mean time you might apply compost (1 cubic yard per 1,000 square feet). You might also spray the yard with molasses at 4 ounces per gallon every other week. Bacteria love molasses.

While molasses will encourage soil bacteria to work harder and faster and is a good food source for them they do need to be present for the molasses to do anything and that requires organic matter in the soil which is what the soil bacteria primarily feed on.

Good point kimmsr. Compost is great for bringing the microbes to the party. You don't need very much. 1 cubic yard per 1,000 square feet is good. Or look into compost tea which brings the microbes with smaller amounts of compost.