Sharon, thanks for posting. There is no reason to connect Elijah Adams (d. 1798 Bedford Co., PA) with the John Adams of New Jersey (born Antrim). This John apparently had a son Samuel who went to Westmoreland Co., PA, but there is no reason to connect them with the Bedford County bunch.
The Elijah Adams who died in 1863 in Bedford was born around 1773, not 1752. He may be the Elijah who was the son of Elijah (d. 1798), but I tend to think not, as this Elijah was a merchant and most likely would have been involved in the administration of the elder Elijah's will---but he was not. This probably indicates that Elijah's son Elijah lived elsewhere (as did his son Robert). Instead, a son-in-law Alexander McGregor, who did live nearby, was an executor.
Curiously, the Joseph Adams of Chester Co., PA, I'm familiar with, as that family intermarried with some another unrelated family I'm interested in, but I don't think he had any connection to the Bedford County Adams.