We need to remember that the Elizabethan age ended with the death of Elizabeth I of England, which occurred in 1603. Shakespeare actually lived on until 1616, and so he wrote a number of his final plays during the reign of King James I of England. Macbeth, from what we can gather, was written somewhere between 1603 and 1607, and many critics argue it was a play that was written in praise of King James I, who was also King James VI of Scotland. He was the first monarch to unite the two kingdoms together under one crown. As King James believed himself to be descended from Banquo, this play is seen as a kind of celebration of the Stuart ascension to the throne. So, this play was very definitely written in the Jacobean era, and can be seen as a play that commemorates the rise to power of King James with his Scottish ancestry.
I think Shakespear wrote all of his books during the elizabethan are.