Viscoelastic hydrogels are gaining interest as they possess necessary requirements for bioprinting and injectability. By means of reversible, dynamic covalent bonds, it is possible to achieve features that recapitulate the dynamic character of the extracellular matrix. Dually cross-linked and double-network (DN) hydrogels seem to be ideal for the design of novel biomaterials and bioinks, as a wide range of properties required for mimicking advanced and complex tissues can be achieved. In this study, we investigated the fabrication of chondroitin sulfate/hyaluronic acid (CS/HA)-based DN hydrogels, in which two networks are interpenetrated and cross-linked with the dynamic covalent bonds of very different lifetimes. Namely, Diels-Alder adducts (between methylfuran and maleimide) and hydrazone bonds (between aldehyde and hydrazide) were chosen as cross-links, leading to viscoelastic hydrogels. Furthermore, we show that viscoelasticity and the dynamic character of the resulting hydrogels could be tuned by changing the composition, that is, the ratio between the two types of cross-links. Also, due to a very dynamic nature and short lifetime of hydrazone cross-links (∼800 s), the DN hydrogel is easily processable (e.g., injectable) in the first stages of gelation, allowing the material to be used in extrusion-based 3D printing. The more long-lasting and robust Diels-Alder cross-links are responsible for giving the network enhanced mechanical strength and structural stability. Being highly charged and hydrophilic, the cross-linked CS and HA enable a high swelling capacity (maximum swelling ratio ranging from 6 to 12), which upon confinement results in osmotically stiffened constructs, able to mimic the mechanical properties of cartilage tissue, with the equilibrium moduli ranging from 0.3 to 0.5 MPa. Moreover, the mesenchymal stromal cells were viable in the presence of the hydrogels, and the effect of the degradation products on the macrophages suggests their safe use for further translational applications. The DN hydrogels with dynamic covalent cross-links hold great potential for the development of novel smart and tunable viscoelastic materials to be used as biomaterial inks or bioinks in bioprinting and regenerative medicine.