1856 can be a little early for consistent civil vital records, so you may have difficulty finding something at the courthouse. Do you know how he was adopted, privately or from an orphanage? A lot of orphanages were run by religious charities, so that may give you clues for looking at Watts families in Rhode Island who were the same religion as the orphanage. (If you're going to look into orphanages, it might be a good idea to start with whatever religion John and Ellen were, because people tended to want children who already shared their beliefs.) If Henry was adopted as an older child, he may have been baptized before going to live with the Jordans. Try church records to see if a plausible Henry exists and work from there.
Something else to consider is if either John or Ellen had roots in Rhode Island. Maybe one had a sister who married a Watts or a cousin whose last name was Watts, and they both passed away and John and Ellen took their son in. Or John or Ellen had a sister who became pregnant by a Watts and had a child out of wedlock. This sort of informal adoption would not leave records.