Florence’s driving rains are forcing hydroelectric dam operators to run generators at full tilt and open floodgates that haven’t been used in more than 20 years.

The Tennessee Valley Authority—which operates dams in Tennessee, Georgia, and other states—began increasing output last week to make room in reservoirs. Duke Energy Corp. has been doing the same in North and South Carolina.

TVA, a corporate agency of the federal government, went beyond cranking up generators and opened spill gates at the Cherokee Dam on the Holston River near Knoxville, Tenn., that haven’t been used for flood control since 1994, spokesman Travis...