I hope it was helpful. I'd say it's very likely that your Augusta was a Brewer. If she married your Murrell in Hawkins/Hancock, she's probably linked to Old Ambrose and his wife, Mary Oma Richardson, although I've never seen that name in my Brewer research. Doesn't mean much, since women--especially if they married and moved away--often got lost in the shuffle. Ambrose's sons (at least, the known ones) were Joab, Frederick, Howell Cobb, Prettyman, Ambrose Jr., Millenton, and Pleasant. There's also a rumored William.
My line comes down through Millenton and, as I said, I've haven't come across an Augusta. This line is fairly well researched,as is the Howell Cobb Brewer line. You might want to start with some of the other brothers.
I checked my copy of Bowman/Congelton,assumed from the date of the first child that Augusta was probably 15-20 when she had that child, and looked for birthdates around 1812-17 and gleaned a little information that may help you. Keep in mind, this is older research. I know the Prettyman researchers have made a lot of progress, and others may have, as well.
Frederick: According to census records, Frederick had four daughters. Only one, Lydia, is named. That leaves three unaccounted for.
Joab: This line appears to be virtually unresearched. Children are "listed with reluctance."
Prettyman: Again, not much research. Census records suggest four girls, only two are named: Elizabeth and Nancy.
Ambrose Jr.: Nothing in B/C, except space for seven children. Not sure whether that means seven children suspected, or are just there as space savers.
Pleasant: Bordering on too young to be Augusta's father, but it's possible. There's a list of 12 children, but no dates.