With the continuous increase of IoT applications, their effective scheduling in edge and cloud computing has become a critical challenge. The inherent dynamism and stochastic characteristics of edge and cloud computing, along with IoT applications, necessitate solutions that are highly adaptive. Currently, several centralized Deep Reinforcement Learning (DRL) techniques are adapted to address the scheduling problem. However, they require a large amount of experience and training time to reach a suitable solution. Moreover, many IoT applications contain multiple interdependent tasks, imposing additional constraints on the scheduling problem. To overcome these challenges, we propose a Transformer-enhanced Distributed DRL scheduling technique, called TF-DDRL, to adaptively schedule heterogeneous IoT applications. This technique follows the Actor-Critic architecture, scales efficiently to multiple distributed servers, and employs an off-policy correction method to stabilize the training process. In addition, Prioritized Experience Replay (PER) and Transformer techniques are introduced to reduce exploration costs and capture long-term dependencies for faster convergence. Extensive results of practical experiments show that TF-DDRL, compared to its counterparts, significantly reduces response time, energy consumption, monetary cost, and weighted cost by up to 60%, 51%, 56%, and 58%, respectively.