I am researching the same people. My husbands 3xgrgrandfather was John/Abraham/Abrams who married Catherine Harper/Harpel b. 1806. I have no certainty about the b of John/Abraham/Abrams and only the info US. However, your idea about researching Josiah is very good. John Abrams is found in the HoHo of one of his sons, Josiah Pittsburgh Tp., in the 1881 census where it says he is 80 years old. His own HoHo as of the 1851 census was: himself 52 born US; Catherine 46; George 21; John Jr. 19; Jonathan 16; Josiah 14; Jane 11; Margret 8 or 5; of course it is not impossible there were older children or especially girls who had already married by 1851.
By the way I have Catherine as Staker, a UEL family; more info available.
Would love to see results of your Long Island searches. Somewhere I have seen: Lawrence, south shore of Nassau Co., Long Island. By the way, since John/Abraham/Abrams and family are often listed as Dutch, settlement in Nassau Co sounds appropriate.
Thanks, Inger Abrams