Too much shade. Can anything be pruned higher, or is it all from neighbors?

That would be a classic shade problem if it wasn't St Augustine. St Aug can take that shade all day long.

I suspect it is a classic watering too frequently problem - especially in the shade, but I could be proved wrong by the OP's reply to this. Here's how you should be watering.

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.

Having said that I had exactly that much shade on my St Aug, and I never ever watered a full inch per week. We got by on 5 hours or less (5/8 inch for me) even in the heat of summer. Now in the full sun in Bandera with temps in the 90s we're only watering once every 2 weeks since the Memorial Day flooding.

Rather than hitting it with compost I would apply ordinary corn meal at a rate of 20 pounds per 1,000 square feet to the entire yard. You said 15,000 square feet but I suspect it is more like 1,500. Or maybe your entire lot is 1/4 acre then you subtract the house, paved areas, and other garden areas that are not grass. You can find corn meal at your local feed store for something under $10 for a 50-pound bag. If they quote you more than $30, then they are talking about corn gluten meal and not corn meal. I would reapply in a month. Corn meal is much more effective than compost as a fertilizer, is a lot easier to handle, and costs about 1/15 as much as compost per 1,000 square feet.

St Aug will spread much faster as the temps cool down into the 70s.

Your soil test is wrong. Your soil pH is 8.0. All the soil between Waco and Floresville and out to Mexico is 8.0. That doesn't matter, though, because St Aug will thrive in 8.0+ soil.

You don't need to aerate. The grass will fill in just fine once you get your watering figured out.

Regarding your shade, it might be nicer to thin out the oak tree. The grass will notice, but it will never be like it would be in full sun. That stuff gets almost too dense to mow in full sun.

Have pruned branches up about 10-20 feet on all the trees and the yard does get partial sun in the mornings (so it's not all crazy). Yard really is 15,000 square feet (1/3 acre subtracting the house). Has been a very wet summer here, so overwatering might be the issue.

So, is there something else that I can plant that will thrive in the shade and eventually choke out the St. Augustine?

St. Augustine is pretty good in the shade. My shaded SA is thinner than my full sun SA, but it is a lush green and hardly any weeds at all grow there in the shade.

There are no turf grasses that will choke out St Aug in the shade. It chokes out almost every grass in sun or shade. There are some weedy bunch grasses that can push it out. Monkey grass (not a true grass) will invade St Aug in sun or shade and take out St Augustine - sometime before the next Ice Age. Monkey grass spreads slower than glaciers move (literally).

St Aug will send out stolons that spread across the surface from 10 to 15 feet or more in a year. That can be hard for other plants to compete with. Bermuda spreads about that fast, but cannot invade into St Augustine because of the shade cast by the coarse blades of the St Aug.