Here is another picture

When did this sod go down?
Did the sod sit on the pallet more than a day?
What has been your watering schedule since that day? How often and for how long?
Is the grass knitted to the underlying soil/sand? Can you lift up any of the grass?

This is almost certainly a disease issue caused by too much water and/or smothering on the pallet. New St Aug is just a little sensitive to it especially if it sat on the pallet too long.

This time of year it is too hot to apply chemical fungicides. Besides that I've never (ever) seen one that worked for St Augustine. On the other hand, I have been very successful getting rid of disease in St Aug using ordinary corn meal. I've used it almost every year since 2002 with excellent results. Corn meal is the material which turned me away from chemicals in the garden.

You can find corn meal at your local feed store, but call first to see what the price is. A 50-pound bag should be about $10. Apply by hand or any spreader at a rate of 20 pounds per 1,000 square feet. It will take a full 3 weeks before you see the improvement. Then you'll see new grass coming up in the brown areas and no lesions on the older grass blades.

Corn meal works through a biological process. It attracts a predatory fungus which feeds on the disease fungus.

Once the sod is knitted to the ground, you can start backing off on watering frequency and going up on the time. Ultimately you should be watering no more than once a week in the FL summer heat and humidity. The rest of the year it would be more like once a month. For now, once you apply the corn meal, I would mist over the corn just enough to wash it down off the grass blades and onto the soil. Then let don't water for several days. The next time you water set out some cat food or tuna cans and time how long it takes your sprinkler to fill the cans. That is your target time for all future watering. That time will likely be between 20 minutes and 8 hours. Every sprinkler and hose system is different. Then that watering with the cans should last for several days. Watch the grass for any signs of wilting. As soon as you see it wilting, water again for your target time. Do this a few times and you'll soon be watering only once every week or so.

I realize that this watering schedule goes against 100% of all the people you talk to in South FL, but it works. As an example I have a house on the edge of the Texas desert. There's about 23,000 square feet of grass. So far this year only about 6,000 square feet has looked wilted. I watered that, but as it turns out, I probably did not need to. My St Aug is a lot more mature than yours, though. The point is if I have not needed much water in the desert, then you should not need much water in the tropics.