According to Sanford Hoadley Cobb's book The rise of religious liberty in America: a history, the answer is no, they did not. See the link.
Church of England.
Yes but to a certain extent. Puritans came to New England in search of Religious Freedom but then only accepted theyir own religion and kicked out any others. Other religious groups often seeked refuge in Rhode Island after being prosecuted in New England.
Did they have freedom offered
For religious freedom
While they were mostly Church of England it was not founded by a religious group or for religious motivations.
what was life like in New Hampshire in 1623?
They were trying to establish a fishing industry.
Primarily fishermen.
For religious freedom
New Hampshire was founded in 1623 and became a Royal Colony in 1679
I'm pretty sure New Hampshire never had an official religion, especially before 1623.
1623
1623
In 1623.
New Hampshire and Maine.
While they were mostly Church of England it was not founded by a religious group or for religious motivations.
In 1623.
John Mason founded New Hampshire in 1623.
what was life like in New Hampshire in 1623?
In 1623 that was Rye.