Electricity grid capacity is rapidly becoming a major bottleneck in the large-scale integration of electric vehicles (EVs). With this motivation, we address a new variant of the EV routing and charging scheduling problem that incorporates a capacitated power grid and adjustable charging power. The problem consists in determining a set of minimum-cost EV routes during a given day and the preceding overnight on-site charging schedule. Routes must satisfy load, time window, and battery capacity constraints, while charging schedules must respect charger availability, grid capacity, and non-preemptive charging constraints. Piecewise-linear recharging functions and alternative travel paths between locations (e.g., minimizing travel cost or energy consumption) are also considered. We develop an exact branch-price-cut-and-check algorithm in which the master problem enforces customer covering, charger availability, and grid capacity, and a backward labeling algorithm generates columns integrating a route and an adjustable-power charging schedule. A dedicated checking problem verifies whether a set of routes admits a feasible charging schedule, enabling early fathoming of branch-and-bound nodes or the generation of feasibility cuts. Computational experiments show that our method solves instances with up to 50 customers and three charging power levels to optimality. Adjustable charging power can significantly reduce routing costs and allow the construction of feasible charging schedules in instances where none exist under fixed power, at the expense of increased computational effort. Our experiments also demonstrate the computational benefits of the checking procedure and the integrated route-and-schedule column representation.