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Teresa, you've got the right family, at least. OBW was a step=nephew to Both Oliver and David. His mother Nancy (named Taylor for her father--Grandmother Mary Hathaway had at least three husbands--Joshua Badger, Joseph Taylor and William Nelson and children by each of them). Nancy Taylor, born in about 1781 at Redstone Fort in Pennsylvania, married a man named Williams in about 1800, who died before 1820, in Montgomery County, KY. Of the known children were Oliver Badger Williams (named for one of his uncles), David, Nancy, Elizabeth, and Margaret, all born before 1815. OBW was born in 1807, and married Mariah Allen, a daughter of William Allen and Catherine Jones "Caty" Allen. She died early in 1859 of TB (consumption) and OBW remarried in Putnam County, IN. OBW and Mariah Allen Williams were my GGGrandparents.
Do we have a common ancestor and do you have knowledge about any of these persons? I have pictures of the gravestones of both David Badger and Oliver Badger Williams in Putnam County and can furnish them to you by E-mail if you send me your address privately.