Three-dimensional (3D) bioprinting has been attractive for tissue and organ regeneration with the possibility of constructing biologically functional structures useful in many biomedical applications. Autonomous healing of hydrogels composed of oxidized hyaluronate (OHA), glycol chitosan (GC), and adipic acid dihydrazide (ADH) was achieved after damage. Interestingly, the addition of alginate (ALG) to the OHA/GC/ADH self-healing hydrogels was useful for the dual cross-linking system, which enhanced the structural stability of the gels without the loss of their self-healing capability. Various characteristics of OHA/GC/ADH/ALG hydrogels, including viscoelastic properties, cytotoxicity, and 3D printability, were investigated. Additionally, potential applications of 3D bioprinting of OHA/GC/ADH/ALG hydrogels for cartilage regeneration were investigated in vitro. This hydrogel system may have potential for bioprinting of a custom-made scaffold in various tissue engineering applications.