The soil here is acidic. The one thing the company that sold us our St Augustine sod said to do was apply lime. I have Lime-Rite Pelletized Dolomitic Lime. WHEN would be the best time to apply it?
There really isn't a "bad" time to put out lime providing you soil needs it. Of course it will need some rain to wash it in so applying it just before a rain shower would be ideal but anytime will be fine.
Send a soil test off to Texas A&M first to find out how much you need to adjust your pH if at all. St. Augustine can grow well in a wide range of pH 5.0 to 8.0. I would aerify and apply lime early spring or early fall. Aerifying allows you to incorporate the lime into the soil at least a couple of inches like moves very slowly in the soil only one inch a year.
If you want more bang for your soil test buck, send the test to Logan Labs in Ohio. You get better (more reliable) results and a lot more for the money. For $20 at LL you get what it would cost you roughly $60 at TAMU. And be sure to tell them, wherever you send it, that you are growing St Augustine.
How are you watering (frequency and duration)?
How high/low are you mowing?
These are important to success with St Augustine.