Tomatoes, squash and beans need a minimum of six hours of full sun per day. In my Mississippi climate, they appreciate some shade in the late afternoon, but that's it.

Lettuces do well in the cooler parts of the year: right now until mid May to early June, and again in the fall and winter (October on). I have grown lettuces in pots with great success. They need a minimum of five hours of full sun per day and more is better.

Have you thought of transforming your shade garden to a perennial shade garden? My favorite bed in my yard is in the shade. Among other things, I grow: Farfugiums, hellebores, Solomons Seal, Japanese Painted Ferns, Arborvitae Ferns, Evergold Carex Sedges, nandinas, Oak leaf hydrangeas, shade loving bulbs, and more. Hostas do not do well, but the Farfugiums are a great substitute. If you like these ideas, check out the perennials forum.

I actually think tomatoes will do fine in such a situation. It may not produce 100% but you will still get plenty if you stick with cherry or other varieties that are prolific to begin with. You could also try growing hot peppers such as habanero.

As far as lettuce goes, try Summer Crisp (French Crisp or Batavia). They can tolerate heat much better. For other greens, you can probably grow chard very well.

"Sun in the morning and early afternoon" probably IS six hours... you can grow just about anything with that much sun. You might have higher yields if it was all day, but you should still do alright. The plants might actually appreciate some shade in mid-afternoon in your hot climate.