Accurate identification of equilibrium points and bifurcation boundaries is fundamental to understanding the nonlinear lateral dynamics of rear-wheel-drive (RWD) vehicles, particularly during aggressive maneuvers such as drifting. Existing studies predominantly adopt simplified two-state formulations or neglect the influence of longitudinal acceleration on normal load transfer, which can produce qualitatively incorrect stability predictions. This work addresses these limitations by developing a comprehensive three-state bicycle model — retaining longitudinal velocity, lateral velocity, and yaw rate as independent state variables — and coupling it with the Modified Elliptical Method (MEM) for combined-slip tire forces. A hybrid computational framework is proposed that combines a Genetic Algorithm (GA) for global search with the Newton–Raphson method for high-precision local refinement, enabling robust identification of all equilibrium branches across the full three-dimensional state space. The implicit structure of the equations of motion, arising from the appearance of longitudinal acceleration in the normal load expressions, is explicitly accounted for through a corrected Jacobian linearization, yielding stability predictions that differ fundamentally from those of reduced-order approaches. Codimension-one bifurcation analysis is then performed by augmenting the equilibrium conditions with algebraic bifurcation criteria, pinpointing one Hopf point and five saddle-node (fold) bifurcation points to ten or more significant figures for a fixed front steering angle of δf = 0.1 rad with varying rear slip ratio sr. The results reveal new equilibrium branches and dynamic transitions not previously reported, and provide quantitative stability boundaries that can directly inform the design of advanced traction-control and drift-stabilization systems for nonlinear automotive applications.