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My mom recently did an ancestry.com search and we came up with alot. We have exactly what you have except I am related to his son John Barker. William Barker is my 16th great grandfather. Just wanted to show you.
The ancestry of James Barker of Rhode Island
was published in 2015 and 2016.
Sources are cited here:
His ancestors were from Essex and Suffolk.
There is no connection to the family in Shropshire.