Digital twin-aided (DT) edge computing is investigated, where users utilize grant-free random access with adaptive rate to offload their tasks to the edge server. A novel, with lower implementation complexity, probabilistic partial offloading scheme is introduced, while each device is assumed to have an infinite buffer to store its tasks. The aim of the proposed work is to minimize the average delay of the partial offloading. To that end, the average delay of waiting in the queue, the delay of offloading, and the local computation delay are extracted by using queuing theory tools. Then, the non-convex problem of minimizing the average delay of all clients is formulated, while taking into account DT imperfections. Successive convex approximation (SCA), alternating optimization (AO), and various algebraic manipulations are utilized to transform the problem into an equivalent convex problem with tractable solution. Finally, simulation results showcase the value of the proposed analysis and offer important insights for the proposed DT-aided edge network. Specifically, the proposed partial offloading scheme is shown to be more delay efficient compared to both local computing and full offloading, particularly, for greater task generation rates at the users. Also, the impact of the DT imperfections at the average delay is shown to be more notable as the number of users, or the tasks’ size, increases.