" next to the sidewalk...."

Ding. There's warm season grass and then there's stinking hot areas, which is what areas next to the sidewalk tend to be. :-) I'll let others chime in who are more familiar, but I think you just used the magic words.

" I already submitted soil samples of different parts of the lawn to get it tested and know what the lawn needs as of fertilizing. "

For the lawn as a whole, you just did a Very Good Thing, although St. Aug isn't exactly what I'd call demanding. For what I suspect is a heat damage issue, it won't necessarily help, but let's wait and see. There are some ways to armor grasses against heat damage a little bit.

You can check to see if the soil is a bit shallow near the sidewalks by driving a long-bladed flat-edged screwdriver into the soil. See how far it goes in. If you hit rock or whatnot, the answer is "quite shallow." If it just stops going in and seems to stick in soil, that's not necessarily a problem (although we might be able to loosen up the soil over the longer-term to assist with water retention).

And there are some other tricks we might manage, depending on how things turn out. Post the report when you get it.

I just got the report today.

Looks like the grass really needs nitrogen but the rest is all right, what fertilizer would you guys recommend?

" what ...would you guys recommend? "

(cough) A decent soil test (/cough)

But seriously, we'll let that one ride for this year, it's good enough, I suppose (I'm not a fan of abbreviated tests that don't give a complete report, and absolutely never a fan of any test that even bothers to report nitrogen, which varies by time of day, moisture in the soil, temperature of the soil, and so on).

At the very least, your pH looks, off the cuff, to be consistent with calcium and magnesium levels, which is more than I can say for some tests I've seen. I'm not a hundred percent happy with your Ca:Mg:K balance, but it's certainly workable and I'm disinclined to fiddle without much better data.

For right now? It's spring in S. Texas, yes? Let's see if we can attract a southern grass person to give you a good, solid recommendation--and I do mean a southern grass expert, not just a stand-in to give you book larnin'. That's very nice, but it's not tuned to your particular weather and projected spring forecast.

I know what I should say, but I'm not directly familiar with exactly what's happening in your neck of the woods right now. It's 44 and raining right now where I'm sitting.

If it goes a couple days, bounce this message and I'll give you a gentle recommendation.