I hunt them in South Carolina. I grew up hunting pheasants and grouse in PA but we don't have pheasant or grouse in SC so I hunt woodcock instead. We do very well and I have very detailed records for the last 16 years. Averaged 18 hunts per year, 36 birds harvested per year, 2.0 bird per hunt average on a bird limit.
Once you know what kind of cover to look for, there should be no reason you shouldn't be able to find enough birds on public land to get your limit. River bottoms with high stem count understory and closed mature canopy is good place to start. Extreme end of cover spectrum would be mature pines with small patches of thick spots within. Could be from where a tree blew down and small pines started to grow or a patch of briars. It doesn't take much of a patch to hold a woodcock. And if you are up for it, sometimes you can find them in rabbit-tat. Short young pines with thick briar understory. They don't like grass ground cover so it the ground cover it grass, go somewhere else. Also, if the river bottom has been flooded out and most of the leaf debris has been stripped away from the flood waters, go somewhere else. One last tip, if you can find these types of cover within 1/4 - 1/2 mile of a plowed or fallow field or fresh clear cut, you will improve your odds of finding woodcock. I lied, here is last tip. Learn to identify woodcock scat. it will take you a long way in helping to find the cover they are using even if the birds are not there when you scout or hunt for them. Best of luck. john