Operator learning has emerged as a promising tool for accelerating the solution of partial differential equations (PDEs). The Deep Operator Networks (DeepONets) represent a pioneering framework in this area: the "vanilla" DeepONet is valued for its simplicity and efficiency, while the modified DeepONet achieves higher accuracy at the cost of increased training time. In this work, we propose a series of Transformer-inspired DeepONet variants that introduce bidirectional cross-conditioning between the branch and trunk networks in DeepONet. Query-point information is injected into the branch network and input-function information into the trunk network, enabling dynamic dependencies while preserving the simplicity and efficiency of the "vanilla" DeepONet in a non-intrusive manner. Experiments on four PDE benchmarks -- advection, diffusion-reaction, Burgers', and Korteweg-de Vries equations -- show that for each case, there exists a variant that matches or surpasses the accuracy of the modified DeepONet while offering improved training efficiency. Moreover, the best-performing variant for each equation aligns naturally with the equation's underlying characteristics, suggesting that the effectiveness of cross-conditioning depends on the characteristics of the equation and its underlying physics. To ensure robustness, we validate the effectiveness of our variants through a range of rigorous statistical analyses, among them the Wilcoxon Two One-Sided Test, Glass's Delta, and Spearman's rank correlation.