Do people in Baton Rouge, Louisiana have southern accents?

There is no one "southern accent". Accents are different in different states and different parts of the state. Some people in east Tennessee and north Georgia have an Appalachian accent. People in south Georgia and north Florida share a very distinctive accent. In Louisiana, settlement by French-speaking immigrants from Canada, combined with local accents to form a very distinctive accent. Calling it "southern" is like claiming that everyone in England sounds the same. 

No, they have the special "Louisiana" accent.
It is more French and sickly sweet.

Gypsyfish is correct and there is not a (singular) "southern" accent.  

https://www.youtube.com/watch?v=VQWPZBmy9eM

If you mean the Georgia honeysuckle accent like in "Gone With the Wind" (ah do declarah!) then no, you won't hear that in Louisiana unless the individual is from Georgia (maybe South Carolina or southern Alabama).

https://www.youtube.com/watch?v=FZ7r2OVu1ssMost White people in Baton Rouge sound a lot like the presenters on the TV news:https://www.wafb.com/live/

Some people have a "country-ish" accent that is harder than the accents in the WAFB video.  It is a lot like the "Redneck" accent Billy Anderson does in his video. 

Many Black people in Baton Rouge have an accent like Mammy in the above Gone with the Wind video. 

I am from New Orleans and the stereotypical accent for NOLA is like a Brooklyn accent.  It is because while NYC was the # 1 port of immigration for the USA in the 1800s New Orleans was # 2. That meant New Orleans got all the same nationalities and ethnicities as NYC, with the same resulting mix of accents. 

I sound like I am from Manhattan except I don't talk that fast or use um and ah.

https://www.youtube.com/watch?v=1hrA9-6o4tI