My mother is Blanche Bonnibelle Rainbow, and I am the GGGG grandson of a John Rainbow (born in 1781, England), who immigrated to the Lincklaen / Cortland area of NYS sometime before 1820.
He was married to a woman named "Avis". Avis outlived John and last appears living on the 1860 US Federal Census. John most likely died between 1850 and 59. John and Avis couldn't read and write, and a critical 1810 census is missing from Cortland Country, so there is a big mystery about who this John Rainbow (b. 1781) is descended from. BTW, this is, I believe, not the same John Rainbow who was married to a Sarah Hunt.
Anyhow, I'd be happy to exchange data with you. I'm currently researching the Rainbows, Bakers, Jacobes, Clarks, etc in my family line. The immediate goal is to fill in the DeRuyter Rainbows, which I've pretty much done. The dream is to bridge John Rainbow to his family in England.