I live in Texas, my office is in Louisiana, work offshore 120 miles south off the coast of Louisiana. I claimed exempt from LA income tax, now it says that I owe LA tax.

Please make sure you answered No to the question "Did you make money in another state?"  You can review your answer to this question by going through your entries under the Personal tab in TurboTax.  

My office is based out of Louisiana though. I just want to make sure LA won't come back and take me to court over not paying taxes. Is this really all that needs to be done?

That doesn't matter.  What matters is where you physically work.  As your place of work is well beyond the territorial waters of the state, you are not subject to Louisiana income tax.