I see you are in Central Arkansas, which means you need a warm season grass. I think bermuda grass is by far your best choice.
Since you already have some bermuda grass out there, it is probably "Common" bermuda. It's not the best bermuda for a lawn or athletic field, but I believe it is a good fit for your situation. Common bermuda seed is the least expensive and I recommend about 100 lbs per acre.
Bermuda grass is best planted NOW. Bermuda seed is small enough that you can just broadcast it on top of the ground, and pray for rain. It would be better if you could cover it somehow with soil, but it must be very shallow because of the small seed size. Depending on what equipment you have, you can probably just broadcast it and drag something over it to help make better seed to soil contact.
I have been successful planting seed as late as September, but if you have a cold winter, it may not be grown enough to survive. It must have enough underground roots established in order to survive a cold winter. It will take a full summer for underground roots to establish.
There are lots of herbicides you can use to clean up bermuda. I have been successful using glyphosate (Roundup) on bermuda to kill some of the grassy weeds. But, you must be very accurate with the rate, and you must be ready to tolerate some injury. For nutsedge, I like to use sulfosulfuron (Outridder). There are others that work also. For broadleaf weeds, I use metsulfuron (MSM Turf and others). Metsulfuron is safer than 2,4-D and it is cheap.
As
@WinterDeere said, you want to get your pH right. Bermuda does best at around 6.0 to 7.0. The only way to know your pH is to have it tested. I suggest you contact your local county agent and get them involved. They can help you with a soil test, and they can recommend herbicides, fertilizers, etc. And they generally give free and unbiased advice, since they are not selling any products.