I can assure you that there is absolutely no connection between the Thomas Moore you say was born at Southwell, Nottinghamshire, in 1574 and the family/descendants of Sir Thomas More (not Moore). This is a pure fiction.
I believe you (and others) have somewhere along the line misread Southwell, Nottinghamshire, for Southwold, Suffolk. This is the town from where the Thomas Moore related to the Youngs family emigrated to America.
Thomas Moore's birth date is a guess, lacking any documentary evidence.