I had a vegetable garden overrun with witch grass once. I didn't know what it was at the time and used a rototiller on it. Big mistake. Made it much worse. It got to the point that I gave up on the garden for two years. Then after research, I decided to use a large piece of clear plastic leftover from an ice rink we had n the yard, and covered the entire area that was about 25x25 feet, with the plastic, weighed it down with bricks and left it there for a whole year. It worked great. We were able to reclaim that area and I got rid of about 90% of the witch grass. I don't know anything about bermuda grass, but I'd research ways of getting rid of that. I can't see using a chemical to get rid of it, in a 50x60ft area where you are going to grow food. If you can't do that, I would think thick cardboard with heavy mulch would be your next best bet. And it will improve the soil and increase your earthworms.

The best way to start a new lawn in the south is with sod. St Augustine seed is not available so it has to be sodded. Hybrid bermuda is only available in sod. There is improved common bermuda seed available but it is still common. If you don't have to have a putting green quality lawn, then bermuda seed will work in the full sun. The problem with mixing bermuda with St Augustine is they each prefer different care. Bermuda needs fertilizer every month and mowing low twice a week in the summer. St Augustine only needs fertilizer 3x per season and likes to be mowed at 4 inches or higher only once a week. If you mow a mixed lawn low it will look weedy because the St Augustine stolon is much thicker than the bermuda stolons. If you mow it high it looks weedy for awhile because the bermuda seed head sticks up before you mow again. St Aug does not need to be watered every week unless you are not watering long enough. If you water it deeply enough, it will grow deep roots that resist the heat. Also when you water deeply and let the surface of the soil dry out, then your weed seeds will not germinate - and that is the subject you asked about. If you just water what you have now, and water it infrequently but deeply, then your weed pressure will fall off. Your lawn will spread because both grasses are the spreading type. If you mulch mow high every week, it will look like St Augustine. If you mow low it will look like bermuda with coarse weeds (St Augustine).

Maybe Senecio vulgaris, common groundsel??? If so, pretty early for bloom although it is a spring bloomer. Can be tough to control as it has developed resistance to many products. Contact your local extension service for the right treatment. hortster Here is a link that might be useful: common groundsel

Before you invest more money for bigger solutions, consider how much time and money you have to actually be handling this. A good lawn is not just a 1-season job of doing something to make it look good. You may have to do something for 2-5 years before you see the result you want. Laying down sod for the entire yard is a fast option but if you aren't taking care of it, you just used a lot of money and will have similar issues again (just like you have described about your previous attempts) A non-grassy solution is that maybe you can turn your entire backyard into a kind of playground or xeriscape area for your kids. Rocks, mulch, pea pebbles, etc -- they will allow for your kids to be able to play in the yard, and for you to be able to spot unwanted plants quickly. Where you do want a particular group of plants, you can do a small bed, either in-ground or in a raised bed. But even at that, it's still something you'll have to maintain throughout the year, because unwanted plants have a way of blowing in through the wind, animals, or even popping up through landscape fabric. If you do have a rototiller already, it's still a good idea to use it to turn the weeds under if you don't want to pull them out first. After that, maintain the area by actively pulling out weeds. You can even make it a game with your kids. Even when I hand-pull weeds, the area I'm working in does get better without high cost of money, but it needs my continuous vigilance to keep it that way.