Erick,
Been to Nassau many times now via cruises.
Pete already covered Greycliff.
As it relates to a LCdLH? Yeah, they are everywhere.
Street guy after street guy, liquor store after liquor store.
Just not what you are expecting.
DON'T dare buy any anywhere there on that island.
Havana Humidor inside the hotel at the Atlantis at Paradise Island is the only legit place to get them and you will PAY dearly. They have a great selection of what I call, "wish I had but hell if I will pay what you want for them".
So, bring your own smokes and kick back and peruse with pleasure and chuckle as we all have done that have been there.