My soil and light are different from yours but I used to play games with the bermuda and St Aug that invaded my stone driveway. One year I would not water and watch the bermuda take over. Then next year I would water once a week and mow it high and watch the St Aug take over. I did that for about 10 years playing them back and forth. Now I'm doing that with some zoysia and bermuda. Infrequent watering and mowing high has kept all the grassy weeds out of the sunniest parts of my lawn for years. I have had the occasional bermuda, nutgrass, and wildflowers (that thing everyone calls wild onion) but they have all gone away. Weeds I cannot control with water and mowing are broadleafed weeds like henbit, oxalis and chickweed. Where do you live?

I have a sloped back yard around my pool, which is much more steep than your slope in the pictures, covered in St. Augustine. I too detest lawns, but I have no money right now, so my St. Augustine is here to stay. I water my St. Augustine with an irrigation system on my allotted days and once yearly treat with weed/feed. Yes, my St. Aug is great on bare feet and very healthy looking. I have no problems with maintaining it on the slope. I'm not bragging, I'm just trying to explain that I have personal experience with this situation. I think your first problem is with the underlying sand. My underlying soil is a thick layer of decayed organic material, on top of a thin layer of sand, on top of a thick layer of loose clay. It's clear from your photos that the mower tires are disturbing the St. Aug's growth. That is very easy to do regardless of the underlying soil, but especially easy in sand such as you have. I think your second problem is with your lawn guy. Seriously. And, I mean absolutely no offense. Your lawn is covered in weeds. It didn't happen overnight, although sometimes it might seem that way. Your lawn guy should know (1) how to mow on a slope without destroying the roots, (2) how to point out a weed problem before it becomes a massive weed problem like you have. How convenient for him that you need sod replacement?!?! Isn't his job routine lawn care, not routine lawn destroying and replacement? That being said, I have a cheap temporary solution for you. I see in the first picture, to the left of the hose/brick path, you have a bush/tree surrounded by a small circle of mulch. And, in the second picture, you have another tree just down slope from the pool, surrounded by another small circle of mulch. I would take a sharp straight-edged shovel and cut away shovel sized squares of sod in a circular pattern around those trees. Making much, much larger circles around the trees. Take a section of the sandy/weedy area, cover thickly in newspaper, wet newspaper, then lay your homemade sod squares on top of the wet paper. Water daily until they take root and can survive on their own. Go back and extend the mulched area around the two trees, so your weed problem doesn't extend under the trees. This should really be done in stages, because the sod squares that you cut out and move need to be kept wet during the moving process. I say my solution is cheap, because you can buy mulch for $1.68 at the big box stores these days, and you would probably need 10 bags. Less than $20. Of course you will have a little extra water expense from having to water daily to establish the newly sodded sections. It is a little bit back breaking, but I'm sure you can either do it or find someone to help. I say my solution is temporary because, ideally, you could eventually plant some sort of hearty, Florida friendly ground cover over that entire area. In the long run it will not require as much water or care as St. Augustine, and you can get rid of the expensive lawn guy! Things to remember: Don't put any weed and feed down on the newly sodded sections either immediately before or after laying the sod. Don't forget to water daily until the new sections have taken hold. Especially during this, our driest time of year. If you choose to use RoundUp on the weedy areas before moving the grass squares, don't lay the grass squares directly on the area treated with RoundUp unless you put in a layer of newspaper or cardboard first. If you can get your hands on some liriope, you could make a border along the brick walk (at the top of the slope), to help hold it all together. Their roots tend to spread out rather than down, so it will keep slope erosion in better check. Lastly, don't let that man with the mower anywhere near the new section. Weed-whacker only until it's established. You probably won't need to mow this area for the next 6 weeks anyway. Maybe someone else will have a better/easier solution, but this is what I did and it works. I certainly welcome any criticism or comments on my suggestion.

The grass is dry. That can happen from not enough moisture or from rotten roots not letting the water up. Are you getting puddles of water in the dry areas? Puddles can allow fungal rot to set in. If no puddles then the areas are not getting enough water. That could be due to sprinklers not adjusted right, runoff from hydrophobic soils, sandy soils allowing moisture to run through, or simply the idea that you need more than 1 inch per week this time of year. My guess is your sprinklers are not adjusted right. That's more of an educated guess because I can see what appears to be a circular shape on the grass in your big picture. Mid way up from center to the right side of the picture is a nice colored circle(ish) green. Outside of that is the dry area. If you put cat food cans perpendicular to the circle and turn on the sprinklers, you will probably see that the cans are not filling evenly.

If you are getting 1" of rain, you should not water. You may be watering too much actually. Normally watering ~5 min 3 times a day is all you need for new sod/seed, without any rain. If your yard is soggy, it's too much and can/will lead to bad things. You just don't want the grass to dry out, the soil should be moist, not soggy. Doing a water volume test will be helpful. Different systems put out greatly different amounts of water in the same time frame. We have posters who can put 1" of water in 10 minutes, others it can take 8 hours. So if you could sit out tuna/cat food can for five minutes and measure your output, that would be great. Some of my popup sprinklers put out 0.25" of water in less than 5 min. I water my new seeds in those area for 1 min and still get too much.