The only info I have is that there were 2 Thomas Meacham's in my family tree. One was born in 1541 and then his grandson born in 1604 (his father was Ambrose born 1578-The first Thomas's son).
Thomas Meacham (1541) married Joanne Pullen 1562 in England
---they had a son Ambrose Meacham born 1578 in England. He married Ursula Perrie.
---they had a son Thomas Meacham born in 1604.
I do not decend from the latter Thomas but from his brother Jeremiah therefore I do not have any further information regarding him at this time.
I hope this helps a little and that perhaps they are the same Thomas you are speaking of.
Lisa