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).

Your links: 2 Those look like they might be 2-3 years old and, being trees, will be hard to remove. Your options are mechanical or chemical. Using chemicals would leave you with stumps. Baby trees are a lot easier to deal with when they first sprout, so be on the lookout for new ones and pull ASAP before they become hard to handle.

I agree with mulching. I worry about extending my garden beds due to the wild violets. We managed to pull the bulk out of the front bed, but it took an entire summer. My in-laws were visiting and bored during the day so would go out and dig down to the roots. They are persistent buggers! I have followed up this year by putting a second layer of thick mulch and taking any foliage off in order to starve the roots. Beside the stray leaf and weeding a lot before the mulch was put down, I have barely needed to weed. Whatever does grow through is weakened by having to push through the mulch. My lawn is almost 100% weeds dispite reseeding every Spring and Fall. I am planning on reseeding the lawn soon for the fall and then placing the feed pellets as an organic preemergent in the Spring. Might help with the garden, too if the lawn stops being one huge weed.... :)