Welcome to lawn care 101. You have some misconceptions that maybe we can straighten out.
First let's look at your soil. Do the jar test to determine whether you have sandy, loamy, or clayey soil. About 90% of people who think they have clay do not. Let's check yours. Get a jar with straight sides. It can be a glass pickle jar or a plastic mayonnaise jar but the sides must be straight. Fill it half way with your soil. Break up any clods you have on the way in. The closer to dust the better. Put a ruler outside the jar, measure it, and take a picture with the ruler. Then put two drops of soap on the soil and fill the jar with water. Cap it and shake the jar until you are satisfied that it is all wet inside. Set the jar down and wait 2 minutes. Take another picture with the ruler. Come back in 2 hours and take another picture. Come back in 2 days and take the last picture. 2 minutes, 2 hours, 2 days. All the sand and rubble will fall to the bottom of the jar immediately (2 minutes). All the loam will fall to the bottom over a period of 2 hours to 2 days. If you have any clay, it will remain suspended in the water for days and days. It will look like brown goo and be far too dense to see through. If you can see any light coming through the water at all after 2 days, then you have almost no clay. The reason you might think you have clay is that some mixes of minerals act like clay in some ways but not at all in others. It can be confusing.
You can probably soften your soil and not have to go through the 10 minute/50 minute drills. Before you get started fixing it, you should go out and see how long it really takes before you get runoff with your sprinklers. While you are at it, take some cat food or tuna cans and measure how long it takes to apply 1 inch of water. These are very important numbers. Some sprinklers can apply a full inch in 15 minutes. If you have one of those, then watering off and on 10 minutes at a time is a waste of water. My sprinkler takes 8 full hours to apply an inch so "obviously" when I need to apply a full inch, I water for 8 full hours. There is no reason you cannot water for hours and hours (and hours and hours). With San Antonio's water restrictions I am limited to 7 hours, but that is puh-LENTY for a normal summer. Last year I needed all 7 hours. Normally I only water 3 hours per week.
So how do you soften your soil? Spray it with soap. I spray 3 ounces per 1,000 square feet of generic baby shampoo from Wal-Mart. I use a hose end sprayer bottle. Measure your lawn, pour in the amount of soap you need, and spray evenly until it is gone. Then turn on your sprinklers to wash it in. Next time you water, skip the soap. The next time after that, repeat the soap and you are done for the season. After that you should be able to water a full inch without runoff.
Now here are the basics of lawn care:
- Water deeply and infrequently. Deeply means at least an inch in every zone, all at once. Infrequently means monthly during the cool months and no more than weekly during the hottest part of summer. If your grass looks dry before the month/week is up, water longer next time. Deep watering grows deep, drought resistant roots. Infrequent watering allows the top layer of soil to dry completely which kills off many shallow rooted weeds and prevent new (weed) seeds from germinating.
- Mulch mow at the highest setting on your mower. Most grasses are the most dense when mowed tall. Bermuda, centipede, and bent grasses are the most dense when mowed at the lowest setting on your mower. Dense grass shades out weeds and uses less water when tall. Dense grass feeds the deep roots you're developing in 1 above.
- Fertilize regularly. I fertilize 4-5 times per year using organic fertilizer. Which fertilizer you use is much less important than numbers 1 and 2 above.
Beyond that, St Augustine will resist the heat much better when it is tall. By tall, I'm talking about knee height. Here is what that looks like.
This lawn is in George West, Texas. My dog is medium size (35 pounds) with a long fluffy tail. That grass has not been mowed since September when I bought the house. It never went dormant in the winter and has not been fertilized with anything for several years. I think it looks pretty good...if you don't need to have a manicured lawn. This tall grass is an experiment of mine. Looking forward to a hot summer.