The sprinklers your little ones are running through are throughout the lawn, yes? I recognize those as regular pop up sprinklers. At a normal spacing and arrangement, they put down an inch of water in 30 minutes. You might need to apply 1.5 inches because it is Texas. If I were having trouble with water penetration, timing would look like 10 minutes on, an hour off, 10 on, and another hour off followed by 10 minutes, another hour to soak in and then the final 15 minutes.
Clay soil that has a compaction/water penetration problem is a candidate for a soil test(pH, Calcium, Magnesium, Sodium levels), and possible usage of a soluble/sprayable Humic formula. The Humic can help re arrange the soil so it is less likely to pack together. A problem in my area is alkaline soil that is the texture of grease when it is wet and hard as a brick when dry. Usage of humic products has helped those soils to not be like grease when wet and to crumble when dry.