The annular geometry with inner cylinder eccentricity and rotation is significant in many thermal and engineering fields, particularly with non-Newtonian fluid flows. A numerical analysis examines the effects of rotation and eccentricity of the inner cylinder on the fluid flow and heat transfer characteristics of shear-thinning non-Newtonian fluids within annular geometry under developing steady laminar flow. The computational model simulates non-Newtonian annular flow using a power-law viscosity model for generalized Reynolds numbers ($100\le Re_{g}\le1000$), flow behavior index ($0.2\le n\le0.8$) and Taylor number $Ta=10^{4}$ with radius ratio $r^{*}=0.5$. The simulation employs hydraulic and thermal boundary conditions, including an adiabatic outer cylinder and a constant temperature at the inner rotating cylinder, while the outer cylinder remains stationary. Results show that axial flow at $n=0.2$ exhibits lower flow resistance and enhances convective transport compared to higher $n=0.8,$ especially for the concentric case $(\epsilon=0)$. However, increasing eccentricity from $\epsilon=0.2$ to $\epsilon=0.6$ alters the heat transfer behavior, with $n=0.8$ yielding the highest Nusselt numbers at $\epsilon=0.6$, due to the strong secondary flows and intensified local acceleration in the narrow gap. These outcomes reveal that heat transfer enhancement is not solely governed by flow resistance but is also influenced by secondary flows, boundary layer stability, and localized acceleration effects.