At the risk of beating a dead horse:
1. Do your trees survive the winter without severe damage? I just want to make sure that the late development of figs is not due to dieback. If it is, you may need some winter protection.
2. You probably want a low / no nitrogen fertilizer -- not the same stuff you'd put on a lawn. Until you have a tree that is the size you want, I'd give a fertilizer with some nitrogen. But once the tree has grown a bit, I'd cut the nitrogen, leaving K, Ph and other minerals. Nitrogen promotes leafy growth but may discourage fruiting. Some manure (slow release N) would probably be OK. As suggested above, fertilize in spring and early summer.
3. You probably want more water, at least until you are sure that you have a well-established root system. I'm imagining that the amount of water required to keep grass green is not enough to saturate the ground and encourage fig trees (or any trees) to develop deep roots. My brother-in-law has a couple of healthy fig trees in red clay in northern GA, but the roots are as thick as my arm and extend at least 15-20' from the trunk; I have to imagine that they penetrate deep into the ground as well. But there's a chicken-and-egg problem -- you have to water enough to get root growth before you can stop watering and let the roots fend for themselves.
4. You may have a late-ripening variety, though that wouldn't explain why your neighbor is successful while you are not. If you add new plants, look for cold-resistant, early-to-mid-season-ripening varieties. The Mt Etnas, as suggested, would work. Also, Ronde de Bordeaux, Florea, Malta Black. I guess that, given your soil, I'd be cautious about southern varieties adapted to / demanding lots of water, such as the LSU figs and Celeste.