The second I think is a clumping fescue. I had the same thing in many places in my yard. This weekend I just completed tearing it all out and planting new. The previous Sunday I sprayed every area with round up to make sure it killed the plant down to the root. Then this weekend I dug up all the grass about 4 inches down and tore it all out. I then filled with topsoil and good dirt and planted new seed and toped that with peat moss to keep it moist and the birds away. Hopefully the new grass comes in and makes it though the summer. I am in ND so our summers are not as hot as other places so I think it will make it. Note this will come back every year if you do not kill it and dig it out.

Floral, I didn't even know there was more than one Aphanes species in the US. The following Weakley snip suggests that A. arvensis has larger leaves. The BONAP distribution maps suggest that the Georgia plants would be A. australis.

It might be nutgrass (aka nutsedge) which spreads by runners but also has bulbs. It is resistant to herbicides, but Ortho makes Nutsedge Killer, and others are Sedgehammer, and Kills Nutsedge by Image.

I'm afraid there are at least four different plants in that clump and it's very difficult to see the details. One if them looks like mouse ear chickweed, Cerastium vulgatum. There is another plant with toothed leaves but it's all tangled up with the others and the flower buds are at the back and impossible to make out. Can you separate the clump out and provide a larger picture?