I'm in your same situation-renting and in TX gardening with a hot shady area. Now I'm sowing (you can direct sow-but I'm not cause I have crickets that eat my seedlings) violets (viola odorada) reportedly handle both our winters and summers in shade and flower. Sounds too good to be true, I'll see. Violas and pansey's also in winter but not summer.

I like herbs so I'm doing herbs in the shade and our heat, these also are evergreen in my winter-salad burnet, thymes-so many, pennyroyal-likes water, lemon balm. These I'm currently growing so I can vouch for them. With herbs buy 4" pots costs about 2-2.50 each. There are many types of thyme, but regular 'plain' thyme can be sown. I believe salad burnet and lemon balm can be sown too.