Answer: You've made a good choice to select an area with full sun. The bermuda can become quite a problem in a garden, as it spreads through above and below ground runners. If you don't get rid of it first, you will be in for a long war!
The organic option for bermuda control is to use the summer sun as your ally. Rototill the area and wait a week to allow the sun to dry out and destroy exposed runners. Then till again and wait a week. This process takes time and will not eliminate the bermuda totally, but can leave you with a managable amount. Choosing this option will mean delaying your gardening until late summer, just in time for a fall garden.
If the garden area is not too large, another option is to till and then pick out all the bermuda you can. Then till again and repeat. Two or three "till and pick" cycles can be done in a week or so, speeding up the process.
The chemical option is to water the bermuda to get it actively growing (very important for optimum control). A few days later, spray with glyphosate herbicide (the most common brand is Roundup) at the full label rate. Adding a tablespoon of dish detergent and 1/4 cup of Ammonium Sulfate fertilizer (21-0-0) to a gallon of spray mix will significantly enhance its effectiveness.
Wait 7 days after spraying to allow the herbicide to move down into the grass plant. Then rototill the area. About 7-10 days after tilling, you can plant with no damage to your seeds or transplants. Whichever option you choose, some bermuda will escape. Never let it up for air! Give it an inch and it will take a mile. So dig up plants as they appear making sure to get as much of the runners as you can.
Your idea of timbered beds is a good one. Make sure the soil mix you add contains plenty of composted organic matter. I have had good success with placing 6 to 10 inches of leaves or shredded branch trimmings between beds for a weed free, all-weather pathway material.
It is a bit late for tomatoes this season. You can plant okra, southern peas, pumpkins, and watermelons now. By July you can begin planting tomatoes for a fall garden as they will fruit in the cooler days of fall. Many other summer veggies like summer squash could go in during mid to late August. Cool season crops like broccoli and cabbage should wait till September. Seeds of spinach, lettuce and carrots will need to wait till October for cooler soil to germinate in.
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