I'm from Louisiana, go to school in Ohio, and did a paid internship in North Carolina. I vote in Ohio and I started working in Ohio in August of 2019. Where did I live?

The answer to all tax questions is "It depends".

It depends on the full meaning of "started working in Ohio in August of 2019".  It depends on your intent. It may depend on whether you are still your parent's dependent.

Did you finish school and started a full time job or just a part time job while gong to school?

Assuming the latter,  you are still a LA resident and must file a LA return. You also need to file OH and NC non resident returns.  LA will give you a credit, or partial credit, for tax paid to NC and OH. 

Assuming school is done,  you are now an OH resident and must file a part year resident OH return and a part year resident LA return, as well as a non resident NC return.  LA (not OH) will give you a credit for the NC tax, since (apparently) the NC income was earned while still a LA resident.