The printability of a photocross-linkable methacrylated gelatin (GelMA) bioink with an extrusion-based 3D bioprinter is highly affected by the polymer concentration and printing temperature. In this work, we developed a gallic acid (GA)-functionalized GelMA ink to improve the printability at room and physiological temperatures and to enable tissue adhesion and antioxidant properties. We introduced a sequential cross-linking approach using catechol-Fe3+ chelation, followed by photocross-linking. The results show that the ink formulation with 0.5% (w/v) Fe3+ in GelMA (30% modification) with 10% GA (GelMA30GA-5Fe) provided the optimum printability, shape fidelity, and structural integrity. The dual network inside the printed constructs significantly enhanced the viscoelastic properties. Printed cylinders were evaluated for their printing accuracy. The printed structures of GelMA30GA-5Fe provided high stability in physiological conditions over a month. In addition, the optimized ink also offered good tissue adhesion and antioxidant property. This catechol-based sequential cross-linking method could be adopted for the fabrication of other single-polymer bioinks.