This is great! You did you homework and you have a lot of "before" pictures.
Those are live oak seedlings. Leave them alone...which is to say, just mow them off and the summer heat will take care of them. You'll see them forever, and the oak will not stop dropping acorns.
The brown spots in the middle of the yard are from lack of fertilizer and possibly some insect damage in the past. The insects caused the damage and the lack of fertilizer is why it has not recovered. I would visit the local feed store and go shopping. I like ordinary corn meal as my primary fertilizer. The cost is 1/6 what you pay for commercially bagged organic fertilizer and it works great. Second on my list (if they don't have corn meal) is alfalfa pellets (also called rabbit food). Apply either of those two materials (or both) at a rate of 10-20 pounds per 1,000 square feet. Depending on your time budget and your hobby budget, you can apply every week, every month, or just once if you want. The more frequently you apply the better the long term results will be. Nothing will happen for the first 3 weeks. That is just the nature of organic fertilizer. But then, all of a sudden, you'll notice the grass turn dark dark green. If you see any yellow spots among the dark green, then you missed those spots. You can go over them again any time. Grain type organic fertilizers are very forgiving. If you accidentally apply 40-80 pounds by mistake, no problem (except maybe to your wallet). I have friends who apply 800-1,000 pounds per 1,000 square feet during the growing season.
The reason for bare soil near the tree is two things. One is there was not enough water applied in the first place. It looks like maybe the sprinkler does not reach back in there. Secondly the tree competes for water. We planted a plant called 'cast iron' around our oak trees. Cast iron doesn't need much of anything and it just grows. Mulch is good, too. Or you can water more deeply when you water. It could be that just changing your watering will help those spots. I recently tested my sprinkler using a cat food can. I absolutely swear by the oscillating type sprinkler for overall sanity. Impact sprinklers miss far too much grass for me and require me to go stand in the yard and water. The oscillator provides a nice every pattern. But anyway I test it and found that in 3.5 hours I got 1/2 inch of water with the sprinkler on a full pattern sweep. So I have been watering at 1/2 sweep for 3.5 hours which should be giving a full inch. That's about right. Check your sprinkler to see how long you should go to get 1 inch of water.
The weed in your picture just above #3...is that more of the oak? I don't think so. You might need to pull that out. St Augustine will choke out most grassy weeds but it will suffer greatly at the hands of clover, dichondra, oxalis, horse herb, henbit, and many other broadleaf weeds.
The last picture shows a problem with wear, shade, watering, and fertilizing. You have a walking path in the shade that probably doesn't get enough care. I would apply compost at a very light rate of 1 cubic yard per 1,000 square feet to all your weak areas to kick start them. For the area you have, don't get an entire yard. You can start with bagged compost. Use enough that you don't cover up the grass. If you focus on the weak spots, that should not be a problem. The side yard might still be slow to recover because of the shade, though. Apply more fertilizer to the weaker areas of the lawn. Again, you cannot do any harm with organic fertilizers unless you smother the grass with them.
On the plus side, I see St Aug runners trying to help you out. All you need is the 1-2-3 of lawn care. The 1-2-3 of lawn care that you apparently found and read here somewhere is in order of importance. Most important is to water deeply and infrequently. Next is mowing high. Then fertilizing. You have watering issues, and with last year's drought, I'm not surprised.
Another thing that might help...if your soil is hard at the surface and water seems to run off, fill your hose end sprayer with either baby shampoo or a non-sanitizing dish soap (cheap ones are not antibacterial). Set the dial for 3 ounces per gallon and spray away. The soapy water will help greatly with water penetration. If you decide sometime in the future to core aerate, do this soap treatment after you water. Water it like you were trying to fill the holes (you can't, but that is the idea). Deeper penetration of the moisture will help in many ways. You can apply the soap weekly for a month and that should be all you need for the season.
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