I'm an hour northwest of Tampa...so not as southern as you, but we have the same struggles with the heat. In our gardens I'm growing bananas and papayas as nurse plants giving shade and wind protection to other growies in addition to their fruit. The bananas and papayas are nice because they grow quickly for some shade and they aren't long lived. Bananas die once you've harvested the fruit and thus far my papayas have lasted about 3 years before they die off. Both plants have several varieties in height. If you can plant those, then a few months later plant some of the other things that may need a bit of relief from the sun.
I also have star fruit, guavas, lemon, lime, lemongrass, sweet potatoes (lovely ground cover and crop - both leaves and potatoes), moringa, passion fruit, ube, chaya, elderberry, pineapples (mine grow best in a bit of shade), Dunstan chestnut, katuk, cranberry hibiscus, june plum, turk's cap hibiscus, mango, mulberry, sugar cane,
root beer plant (hoja santa). Definitely look at the crops from Africa and tropical areas.
....and then there's the annual stuff I grow....beans (green and for drying), okra, herbs (culinary and
medicinal), melons, squash, cucumbers, tomatoes (try to find Everglades tomatoes), corn, peanuts (another wonderful ground cover). My harvests have smaller and inconsistent due to trying to find varieties that will grow with our weather and tweaking the soil. I've had
enough to save seed from plants that have done well though.
For lettuces, I started to use the Kratky method of hydroponics this year. It's so easy to set up....no pumps or anything. I can keep my growing bins on my back porch where they get plenty of light, but are protected from the heat. We've had several hot days so far this year, and things seem to be good. We'll see how we do this summer.
I think if you can establish a food forest of sorts to give some shade and cool the area where you are trying for your kitchen garden, you may have better results, especially going into summer. You can also try erecting something that you can suspend shade cloth from until you get your shade from your plants.
Hope this helps! Very best of luck! 8 )