I live in Florida and have St Augustine grass. Its been very dry!

Hello I also live in Florida. You could go to Home Depot and buy the St. Augustine grass plugs or even sod. It all depends on how much of an area that has dried out. Your grass needs to be watered. I have a different kind of grass - Bermuda Grass..mine recently had an area of where my grass died due to having some construction done. I was told to pull all the dead grass and vines out. It did eventually grow back.

You can save it ~ http://www.popularmechanics.com/home/lawn-garden/how-to/a8014/how-to-save-a-drought-stricken-lawn-12203548/

I would suggest a deep soaking, with root stimulator applied right after. Then your peat moss, mixed with some potting or garden soil: enough to cover most of the SA grass with about an inch total. The green that's already out will probably die, but you are interested in rebuilding the root system. Leave the cover, uncovering about Labor Day. (It's July 16 now.) This will give about six weeks of rebuilding underground. When it's close to the usual date for the first frost, or if it just seems very cold to you, mulch again at least the bare places and your shrubs and young trees. In the spring, uncover the St Augustine and it should do much better. Best wishes 😇

Here's everything you ever wanted to know about St. Augustine, Spoholik:

http://edis.ifas.ufl.edu/lh010

You might have better luck with your bare spots if you pick up some sod squares at your local garden center to fill them in with OR use some sprigs to start new growth. You'll need to keep them watered until established.