Terahertz (THz) highly directional signals are susceptible to beam misalignment, which makes it not easy to simultaneously achieve excellent communication and radar performance in dual-functional radar-communication (DFRC) systems with high-speed mobile users. This paper investigates adaptive beamwidth design over time for THz DFRC systems, where the user moves with time-varying speed, and its angle and distance from the transmitter change frequently with time. Firstly, as beam alignment is a prerequisite for both radar and communication, we develop two performance metrics related to beam alignment probability, i.e., successful detection probability and wireless communication rate, to capture the impact of beamwidth on the performance, respectively. Secondly, we formulate a sum rate maximization problem with respect to beamwidth optimization, subject to a minimum successful detection probability. Thirdly, since both the objective function and constraints are non-convex, we solve the problem by using a simulated annealing algorithm. Numerical results verify the superiority of our work, and the sum rates as well as the transmission windows are larger in a drier atmosphere.