Answer: I can think of three well mannered groundcovers that will grow in the area you describe. Consider Ajuga (Ajuga reptans), grows 3 inches high and spreads to form a thick ground cover. It is not aggressive and produces 6" tall purple flower spikes in the spring. At other times it has green or variegated green/purple foliage. It's evergreen.
Liriope (Liriope muscari) grows taller and flowers in summer. Makes an excellent groundcover for your gardening region. Violet, white or pink flowers.
Mondo Grass (Ophiopogon japonicus) grows 6-12" and produces small lilac flowers. It is especially well adapted to Louisiana and requires no particular maintnenance.
Once established these plants can tolerate occasional flooding during rainstorms and the roots should knit together enough to keep the soil from eroding.
Best wishes with your landscape!
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