I live & work in Louisiana. I will be moving to another state later this year. I have NOT turned in a resignation, nor have I verbally given my supervisor an exact date only it will be October or after. I myself am not sure of the date. I was told by my supervisor that they were going to start looking for my replacement due to the fact that it would take several months for me to train them. I was told with witnesses, I was not being pushed out of my position! On 5-1, I started training my replacement. As of today 5-15, I was told by my supervisor that she would be "turned loose & on her own as of tomorrow" When asked what I would be doing, I got no response.
Any advice would be helpful!
What are my rights if any at all?
I know La. is an at will state. I also know how my supervisors are when they get ready to fire you. They stack the deck against you with write ups so you do not qualify for unemployment.

They may or may not be forcing you out. And maybe they want you to oversee her work for a while. While none of this that occurred is illegal, you should speak with someone who can give you an answer about your future with the company.

FWIW, in the future, giving such a long notification of your intent to resign sounds good in theory, but often ends up backfiring on the employee, no matter how good your intentions. Since the company knows you're leaving at some point, they very well may figure that they should get your replacement while the getting is good; they have no legal obligation to wait.

Having said that, I have an excellent employee who gave me two months' notice of her intent to retire, and I'm working her up to the last minute. ;)