North Carolina annexed Tennessee as a county, Washington. After the Revolution, North Carolina gave Tennessee to the United States, but the Federal government did not want it and so the State of Franklin was then established in 1784 and it lasted until 1788. Tennessee became the Southwest Territory in 1790 and finally the 16th State in 1796.
The first widespread census for Tennessee was the 1820 census. It was also the first one in which our Putman families appeared. Not all county census data remains, so there well may others in 1820 that simply do not have existing records. From 1830 on most of the reports still exist.