Controlled burns can work, however you've got trees close by....and they can also create more problems with other weeds. Some seed simply need heat and or smoke to start germination.... Chemicals are not an option for me, personally around food. Your choices are slim right now - basically hand weed around the plants that are growing, and then mow the areas that don't have food directly in them (yet) if you can, is probably going to be the easiest. You will do fine, you will "get" this thing. Realize that it is simply going to take time, lots of time for you to manage this size of space. My thoughts now are to save what you can by doing what I mentioned above. In the fall, concentrate on a much smaller growing space. Lay cardboard down (it will take a LOT and if you can get free wood chips from tree trimmers, neighbors or anywhere, get them. Spread them on the cardboard and leave it all winter. Let it smother what is below. But for now it seems your ambitions were larger than the amount of time you'd like to dedicate to working in a garden. That is the balance you are going to have to figure out...what you offer the garden in maintenance vs. what you have. There are also spray options if you so choose to do that. You can mow an area now, spray it lightly with kill, leave it in the sun a few days-a week (most chemicals work with sunshine) and then go ahead and mulch REALLY heavily (especially with cardboard and wood chips). Next year plant sunflowers in that spot, the year after you can plant food. (sunflowers are used to clean up toxic areas). You will however always have to watch the area - weed killer is only going kill the current weeds, not the seeds for the next year (or from past years). MULCH will be your answer, and it will make some lovely soil too :)

Sounds like nutgrass. As you've discovered, it's very persistent because of the small bulbs on its root tips. You can win if you remove the green growth before the plant develops 4 leaves, then repeat as needed until the bulblets are starved. In other words, you have a new hobby.

First contact your local office of the Clemson USDA Cooperative Extension Service and the can help identify that and can also help you get a good, reliable soil test which might do as much to control this as anything else. Many "weeds" grow in soils that are not good soils for turf grass and turf grass most often needs a good, healthy soil or it will not grow well. At the same time dig in with these simple soil tests, Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top. 2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up. 3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart. 4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer you soil will smell. 5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy. to learn more about the soil you have and what yo need to do to make that soil into something that will grow a good, healthy turf. Here is a link that might be useful: Clemson CES

Kill everything with roundup and re-seed mid-August. Roundup is safe to apply up to a day before seeding, just make sure you don't get the extended control one, it has pre-emergent in it. Seed needs to be kept moist at all times to get good germination, which means watering 3-4 times a day for 5-10 minutes until germination. With KBG it can take up to 3 weeks. Then you can start cutting back on frequency and increasing quantity a little at a time. I would highly suggest you use either a starter fertilizer with mesotrione(Scotts) or spray Tenacity at seeding time. Apply pre-emergent every spring going forward, and if you can even in the fall to keep winter weeds under control.