Strange. I'm from Houston. It has not been raining very much at my house. It's been blazing hot though and most people's yards are starting to look terrible.

My first thought would be to ask what your watering schedule is. Some people insist on watering every day for whatever reason. Guy across the street from me moved into his new house a couple months ago. He waters his lawn every night. Being a new lawn I think he's getting away with it for now. He will eventually have root rot and a weed and crabgrass infestation.

I try to water once a week for about 45 minutes with a sprinkler. Is that enough or should I increase the watering time?

And is there a good method for repairing these patchy spots? I see Home depot and amazon sell some grass seed mix but I'm not sure that will be a good solution.

You have St. Augustine. You would need sod. So the seed would not do much good.

Try poking it with a flat blade screwdriver. It should go in a few inches. If not, its too dry and you need to increase watering. St. Augustine is thirsty. Also have you put any fertilizer down?