Analysis and computer simulations of structural transformations and viscoelastic response functions are presented for the steady rectilinear simple shear flows of a flow-aligning cholesteric liquid crystal oriented with the helix axis along the vorticity direction of the flow; the helix orientation is known as the fingerprint texture. Computation and analysis of orientation and stress relaxation after cessation of shear are also given. The governing parameter that controls the orientation response to the imposed shear is the Ericksen number E (ratio of viscous to elastic effects). For a sufficiently high value of E, a sheared cholesteric liquid crystal with a fingerprint texture originally aligned along the vorticity axis is found to undergo the helix uncoiling transition and adopt the nematic flow-aligned ordering. On the other hand if the Ericksen number is low, the helix rotates driven by vorticity and the orientation state is characterized by traveling twist waves. The transition between the traveling twist wave mode and the stationary nematic mode is found to be mediated by the expansion of the cholesteric pitch, and occurs at a critical value of E, denoted by E,. Close form solutions for the helix pitch as a function of the Ericksen number and for the critical Ericksen number E, for helix uncoiling are obtained and shown to be in good agreement with the numerical solutions. The viscoelastic modes during shear are identified and analyzed. For steady shearing the classical three-region apparent viscosity curve is obtained. Stress relaxation after cessation of shear is computed and explained in terms of the governing scales of the model and in terms of elastic storage modes that operate during shear. Typical oscillatory underdamped stress relaxation oscillations generally present in liquid crystalline materials are found.