Nigelscott - St Augustine is the most common turf grass in south Louisiana. We have many of the same issues that you do in Jacksonville.
I tried do post the actual publication but my internet is acting up this morning.
https://search.ufl.edu/web/#gsc.tab=0&gs ... as.ufl.edu
Scroll down. There is a link for a St. Augustine Maintenance Calendar. It is specific for your region of Florida.
St. Augustine in south Louisiana does not do well with a weed and feed at this time of the year. Straight fertilization, adequate water and proper mowing practices will go a long way toward improving the health of your lawn and smothering out unwanted weeds.
For a fertilizer do read the package carefully. I read on another link that you need to look for a Florida identification number on the package. The letter F followed by an identification number. The package must also specify St. Augustine on the label.
Proper mowing is a big issue. Most people mow St. Augustine too infrequently and too short. St. Augustine needs a mowing height of 2 1/2" - 3 1/2". Never remove more than 1/3 of the grass stems in a single mowing. That means more frequent mowing. Frequent mowing will also prevent the weeds from going to seed. You have a long growing season so you should see some improvement before it browns in the winter. Follow the calendar for proper care and by this time next summer you should have a beautiful, lush lawn. There really is no over night fix. It takes time and dedication to achieve and maintain a beautiful lawn.
Congratulations on your new home and good luck with your lawn.