On my side of the Alderson tree (Descendants of Thomas son of John)there are a few Jesses. Most likely candidate based on age for me would be Jesse B abt 1802. As far as I know son of Curtiss Alderson and susannah Larue. I dont know any thing about any adoption for this Jesse. Their parents lived long lives and had a few children but the name was more common in this line. There's even a Jesse James Alderson born in 1835.