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First, I agree the LSD is a good source but not always reliable. My information comes from a book "Charles Peckham" I found in the Portland, Oregon Genealogy library. We do indeed share several ancestors, John, John, Isaac, Joseph, (father of Sarah). His son Charles is my next ancestor. I have quite a bit of information on the above if you would like to share your address I will send it to you. Charles also went to Ohio and I am trying to verify that information.
I would love to work with you on any research project. There should be historical and genealogical societies in Middletown, Rhode Island. Also the DAR might have information. Joseph fought in the Revolutionary War. I look forward to hearing from you.