A new cavity scheme for a self-sweeping fiber laser with separated gain and absorption dynamics gratings is presented. The scheme is experimentally realized in an Er-doped ring fiber laser generating in the continuous-wave (CW) regime near the wavelength of 1604 nm. Switching between single longitudinal mode stabilization and wavelength self-sweeping regimes is demonstrated by controlling the intracavity losses in the laser. The pump power and optimization of the intracavity losses made it possible to demonstrate record sweeping range of 2.6 nm in a single-frequency self-sweeping regime in the telecommunication L-band.