The Development of bioresponsive extrudable hydrogels for 3D bioprinting is imperative to address the growing demand for scaffold design as well as efficient and reliable methods of tissue engineering and regenerative medicine. This study proposed genipin (5 mg) cross-linked gelatin (1 to 1.5 g)-hyaluronic acid (0.3 g) hydrogel bioink (20 mL) tailored for 3D bioprinting. The focus is on high cell loading and a less artificial extra-cellular matrix (ECM) effect, as well as exploring their potential applications in tissue engineering. The bioresponsiveness of these hydrogel scaffolds was successfully evaluated at 37 °C and room temperature (at pH 2.5, 7.4, and 9). The rheological and mechanical properties (more than three times) increased with the increase in gelatin content in the hydrogel; however, the hydrogel with the least amount of gelatin showed the best extrusion capability. This optimized hydrogel's high extrusion ability and post-printing shape fidelity were evident from 3D and four-axis printing of complex structures such as hollow tubes, stars, pyramids, and zigzag porous tubular (four-axis) scaffolds (printed at 90 kPa pressure, 70 mm/s speed, 22G needle, fourth axis rotation of 4 rpm). 3 million/mL MC3T3-E1 mouse osteoblast cells were used in preparing 3D bioprinted samples. The in vitro cell culture studies have been carried out in a CO2 incubator (at 37 °C, 5% CO2). In the cytocompatibility study, almost three times more cell viability was observed in 3 days compared to day 1 control, proving the non-toxicity and cell-supportiveness of these hydrogels. High cell viability and cell-to-cell interactions observed at the end of day 3 using this moderately stable hydrogel in 3D bioprinting exhibit high potential for precise cell delivery modes in tissue engineering as well as regenerative medicine.