I believe Rogersville, TN, the present-day county seat of Hawkins County is the most likely place you would find the records of Spencer County, Franklin. Unfortunately, many offical records dating from the troubling years from 1784 to 1788 were lost when both North Carolina and Franklin claimed sovereignty over the same territory.
Depending on the date of your inquiry, another place to look is Blountville, TN. It's the present-day seat of Sullivan County. In 1785, during the short life of Franklin, Spencer County separated from Sullivan County.
Good luck on your quest of information. Wish I could be more helpful.