In this paper, a class of vehicle routing problems (VRPs) is considered in which the vehicle dynamical model depends on the chosen route. Specifically, the mass of each vehicle is determined by the loading demands of the locations that the vehicle had previously visited, which leads to variable dynamical models. Hence, an integrated optimal vehicle routing-and-control problem is formulated. The objective function (total cost) considered is a weighted sum of the traveling time and energy consumption cost of the vehicles. A sequential approach can be employed to solve this problem, in which the routing solution is obtained first, followed by finding optimal control costs of all vehicles. However, the sequential approach may not obtain an overall optimal solution, since the routing solution can affect the control cost and vice versa. To the best of our knowledge, an integrated vehicle routing-and-control problem with load-dependent dynamics has not been considered in the literature. Also, there has been no simultaneous routing-and-control solution approach for such a problem. In this paper, a upper confidence bounds for trees (UCT) approach for solving the proposed problem has been implemented. Two test examples are used to demonstrate the proposed approach: (i) a benchmark VRP from the literature, revised in this paper with an embedded optimal control problem, and (ii) a notional integrated routing-and-control problem formulated for a small area based on the New York City street map. For both examples, the proposed approach obtains an improved solution when compared with the sequential approach.