I too am a direct descendent of Thomas Eames born 1618, and one of the first settlers in Framingham, MA. I have a chart which takes the family down to Elizabeth Emes (1808) in Ontario, Canada.
In the U. S. they seem to have dropped the "E" and in Canada the "A" has been dropped.
Contact me at email@example.com
, and I may have some information you can use.