Large lawn in the Heights? Wow. That's something I don't see after growing up there through high school by 290 and 610. I suppose that Scott's Organic fertilizer from Lowe's or Home Depot would be sufficient at a minimum. Who cares about the weeds? Just mow high (3-4 inches) and you will be fine. That's what we do at my mom's house few miles from you and it's the greenest lawn on the block during the winter and it's hardly ever watered during the summer. What kind of garden? Mulching is the best way to conserve the water and improve the soil dramatically. Look for tree trimmers around Houston and ask if they would dump a load of wood chips. That saves you a few hundreds of dollars. I don't really know where in houston to get compost for free. As far as I'm concerned LETCO (Living Earth Technology) has the best compost in Houston for bulk. It does help a lot. One cubic yard per 1000 sqft is all you need. It's the delivery fee and fuel surcharge that suck so if you have a truck or a trailer with sidewalls, it's great. Fortunately, it's only one time application. All you have to do is feed the lawn with organic fertilizer every 3 months. Nothing to it. There is two organic places in Houston... Wabash feed store off Washington nearby Memorial Park and Southwest Fertilizer off Chimney Rock (i think) outside 610 loop not too far from the Galleria. My preference is soybean meal but I don't know how much they cost there. I think I saw it at Southwest fertilizer but that's over a year ago.

Almost nothing looks good about that lawn. Need a picture taken from 3 inches away from the blades on a cloudy day (or in the shade) without the flash to see if it might be a disease. Try this, dig up one square foot of grass in the bad area and see if there are any grubs. If you see fewer than a dozen grubs, then don't do anything. If you have more than a dozen, at this time of year they have stopped feeding, so killing them would be a waste of time and money. They have done their damage and that would be what you see at the surface. The damage was done months ago. Check your insecticide to see if it works on caterpillars. The best thing for caterpillars is a product called BT Worm Killer. It only kills caterpillars. It takes about 48 hours, but it stops them from feeding immediately.

It's still early. Give it another 3-4 weeks and reevaluate. Improper watering is almost always the culprit with St Aug issues. Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.

Looks like a fungus ring I wouldn't do anything stuff like the tend to fix it's self.or maybe grubs.