Well, you're in the right area; the Atchafalaya Basin has the reputation of being the premier doodle wintering ground. We look for closely packed vertical stems or brier tangles. For us, presence of water seems to be key with scattered puddles which raises the water table and consequently lifting worms closer the forest floor. When we first got serious in pursuit, we first concentrated in areas where there were large areas of rivercane, chest high or higher. We broadened our search to include thickets of huckleberries/sparkeberries, new grown pine clusters and dense colonies of hardwood sapling thickets including sweetgum. Nearness to open fields or wide grassy road shoulders of the same thickets seemed to help. We've found them in mature pine forests but mixed with hardwoods and clusters of cover and nearby puddles. We've broadened our search areas over the years and are constantly searching for new areas. If the cover is easy to walk through, look elsewhere, but we've found enough in the open to suggest otherwise, but odds are, the thicker the better. A shortcut in finding good cover and what to look for is to hire a guide. There are more than a handful in your area, but be sure the guide understands your intentions.
Good luck. Gil
PS: I just saw where you saw some hunters coming from the area and were pretty closed mouth about action. You were in the right spot. Anyone who discloses where he shoots woodcock is talking way too much.