Native collagen doughs were processed using a syringe-based extrusion 3D printer to obtain collagen scaffolds. Before processing, the rheological properties of the doughs were analyzed to determine the optimal 3D printing conditions. Samples showed a high shear-thinning behavior, reported beneficial in the 3D printing process. In addition, tetrahydrocurcumin (THC) was incorporated into the dough formulation and its effect on collagen structure, as well as the resulting scaffold’s suitability for wound healing applications, were assessed. The denaturation peak observed by differential scanning calorimetry (DSC), along with the images of the scaffolds’ surfaces assessed using scanning electron microscopy (SEM), showed that the fibrillar structure of collagen was maintained. These outcomes were correlated with X-ray diffraction (XRD) results, which showed an increase of the lateral packaging of collagen chains was observed in the samples with a THC content up to 4%, while a higher content of THC considerably decreased the structural order of collagen. Furthermore, physical interactions between collagen and THC molecules were observed using Fourier transform infrared (FTIR) spectroscopy. Additionally, all samples showed swelling and a controlled release of THC. These results along with the mucoadhesive properties of collagen suggested the potential of these THC–collagen scaffolds as sustained THC delivery systems.