Hi and welcome to ATP
Let's hope we can help you out here. A picture of the area would be really helpful if you can post one.
There really are a lot of factors that could be contributing to your problems. So here come some more questions for you:
Have you had your soil tested in several different spots along your garden patch? Your County Extension service will do soil testing for you, or you can buy a good soil test kit at Lowe's or Home Depot and do it yourself, which might cost less. If your plants in pots do fine, then fail when you put them in ground, that does indicate a soil problem. Amendments such as plenty of compost are the answer here. Our County landfill makes really great compost and it is free for the fetching.
How old is the 30ft. stucco wall next to the area? Newer stucco can leach alkaline stuff into the soil for a while after it is applied. Also, how old is the structure, and do you know if fill dirt was brought in as a base for the foundation? Sometimes fill dirt is vastly different than the 'native' soil and can be anything from infertile clay to inert sludgey stuff.
The blazing sun for most of the day will really limit what you can grow there, and if you are looking to plant new stuff, fall and winter would be my preferred times, to give new plants a chance to grow some new roots before the hot weather comes back. A plant that says "Full Sun" on the label may or may not be suitable for a really blazing hot area such as you're describing. eg. Roses all say "full sun" but frankly my roses do a lot better with shade through the middle of the day in the summertime. So the full sun designation might work in Michigan or New York state, but full sun in Florida is a whole different ball game. A lot of full sun perennials can't take the sun in Florida.
Once you've checked the soil, I'd be looking at some small to medium sized trees to plant along that wall, along with your other new plantings. Then in the cooler months when the sun angle is lower, the other plants will get more light but in the blazing hot summer months the trees will offer some relief. They'll also lower your cooling bill, once they're growing up a bit, if that 30ft. wall is the wall of your town house.
I'm not sure about the ligustrum question. Will have to go looking for the answer to that. But I do have two good size ligustrums and they do allow some things to grow under them, it seems.