Since its inception, the steganography system (SS) has continually evolved as an indispensable method for covertly concealing sensitive information. To enhance the performance and augment the concealment capacity of traditional SS techniques, the integration of contemporary algorithms, particularly those rooted in artificial intelligence (AI) and deep learning (DL), has emerged as a vital necessity. In light of this demand, we present a novel 3D steganography algorithm in this work. Our proposed algorithm is structured around three main phases, commencing with a preprocessing stage that handles both the secret picture meant for concealment and the 3D mesh structure utilized as a covering media. A mesh traversal approach is presented to defend the 3D stego model from possible vertex reordering attacks. This traversal technique, rooted in breadth-first search, is distinguished by its unique strategy of prioritizing the nearest neighbors during the traversal process. Furthermore, this algorithm ensures the consistent generation of the same traversal order, even when the mesh undergoes transformations like rotation, scaling, or translation. Our steganography approach introduces an innovative difference shifting scheme, notable for its reversibility and incorporation of a blind extraction method. A further advantage lies in the first-time application of a logistic chaotic map to randomly embed secret bits within the mesh 3D model. The experimental outcomes conclusively show that our suggested steganography algorithm successfully hides hidden information within 3D mesh models, limiting distortions in perception to an unnoticeable degree. Surprisingly, the 3D stego model allows the original hidden image to be retrieved. A well-rounded trade-off among distortion and capacity embedding can be accomplished by changing threshold values.