I live in Lafayette, la. We have similar climates but different soil types. I highly recommend having a soil test done both for nutrients and pathogens.
The most popular varieties in Louisiana are Celeste and Southeastern Brown Turkey. Both have a high sugar content and are great for canning. Love fig preserves!
The information in this link should be applicable to you.
https://www.lsuagcenter.com/NR/rdonlyres ... LOWRES.pdf
I do not have a fig in the yard at this time. I have grown figs before and my Mother has several trees in her yard. Our common practice is to fertilize with 8-8-8 mid February and early May.
I use a plastic peanut butter jar with holes punched in the lid. Put your left shoulder against the foliage and walk around the tree fertilizing as you go. 1 round. Water in. If you do not have sufficient rain then water heavily every week. Both varieties ripen around July 4th. Stop watering mid June. Let the sugar concentrate in the fruit.
I really think you have a soil issue. Have it checked out.