Most likely too much shade. How many hours of sunlight does that area receive? If less than 3 hrs, then you have a loosing battle. Is your soil ph lower than 6.5? If not, you don't need to use lime.

I have three live oak trees in the front on my north side and have no trouble getting grass to grow. It might not be the thickest stand of St Aug, but it doesn't die out every winter.

Sand is my favorite St Aug substrate, so no problems with that.

What is your watering regimen? How often do you water and for how long each time. My guess, based on years of reading posts from Floridians with St Aug problems, is that you water far too frequently, but I'd like your confirmation before I launch off into corrective action. I further suspect that your lawn dies out from a fungal disease exacerbated by the watering regimen.

How high/low do you mow? Again, my guess is that you mow too low.

If you are not using organic fertilizer, you should start. The main reason sand works for me is that I've been using organics for 12 years. By organics I mean something like corn meal, alfalfa pellets, soybean meal, or other readily available ground up animal feed that you can find at your local feed store. The application rate for these organics is 15 to 20 pounds per 1,000 square feet. You can apply them any day of the year, or every day of the year, without fear of harming the grass. What these feed types of fertilizers do is develop a healthy family of microbes in your soil. Then the microbes will feed the actual plants and grass growing there.