Jeremy, I hope you have deep pockets as you will have to drive quite a ways to find turkeys from Jeanerette. The closest public areas to you will be Sherburne WMA and Indian Bayou. Both have reasonable turkey populations.
I would suggest using a boat on Sherburne and you will probably need feet or perhaps a bicycle on Indian Bayou. Both are located in the Atchafalaya basin. You can use the LDWF website to get info/maps for Sherburne. Sherburne has some pretty good fishing so you can fish and use a GPS between now and turkey season to create waypoints to likely hunting areas (you will want to have the waypoints so you can get there before daylight). A trip or two in early March on good gobbling mornings should put you in good locations. Get quad maps of the area and carry a compass & GPS so you always know where you are.
Indian Bayou is controlled by the Corp of Engineers. Their office is in Krotz Springs and my information is that those folks are real helpful in pointing you in the right direction.
I know Sherburne has lottery drawings for the first part of the season and the same may be true for Indian Bayou. Even with the lottery draws, expect human competition. It is the nature of the beast for public Louisiana hunting.
Why be good when you can B-Mobile or Spin & Strut.