Travel by ship from Port of Baton Rouge?

Yes. There is a stowaway website that has been discussed.

Decent accommodations on some ships.

Not as cheap as you think.

That's an interesting question. Years ago, the owner of the company my dad worked for used to travel that way out of New Orleans. They would travel to some port in South America and fly back. The guy and his wife were voracious readers and would go through a suitcase full of books while sitting out on the deck. As I recall, there were about 6 cabins available.
Am interested to see what others may know.

this is actually something that I did not know existed...probably not something that I would like to do many times, but it may be interesting a time or two. I think these folks come through NO...

LINK

Just take a cruise and get a balcony room if you don't want to see other people. Better food, I assume and something else to do if you do get bored. There are Transatlantic cruises and repositioning cruises you can take if you want to travel after.

Bro, do you even Huey P. Long?