There was an original sin of man. It is told about more in Morgoth's Ring. When men had just awoke they turned away from the Voice or Eru and worshiped Melkor and did evil things. The Edain were the ones who repented and fled him. Tolkein decided the original fall of Man should remain ambiguous though. The whole mythos is about a set of Falls. The elves fall due to the Kin-slayings of the Bay of Swans, the fall of Dior and the Battle at the Havens. Men fall twice, one the original fall that we aren't supposed to know about and the second fall when Numinor is destroyed. Man did not fall in the same way Adam and Eve
did, but they fell nonetheless. Did this mean they forever had Original Sin? Well they carried the repercussions of it (shorter life, fear of death etc)