Smart hinges fabricated using three-dimensional (3D) printing have been accepted in the aerospace, robotics, and biomedical fields since these devices possess a shape memory characteristic. Shape memory polymers (SMPs) are the preferred materials for creating smart hinges due to their ability to achieve programmable complex geometries. However, fabricating SMPs with embedded components remains a challenge due to the constraints of current 3D printing methods and material limitations. This study investigated the use of a hybrid 3D printing method, direct ink writing (DIW), and embedded 3D printing (e-3DP) to print smart hinges with an embedded circuit to act as a strain sensor. The main components of the SMP included tert-Butyl acrylate (tBA) and aliphatic urethane diacrylate (AUD), but this SMP ink had a low viscosity and could not be used for DIW or e-3DP. Fumed silica (FS) was added to the SMP to tune its rheology, and it was shown that the FS concentration significantly affected the rheological properties, dry-out process, filament geometries, and self-supporting capabilities. This study presents a hybrid 3D printing approach for creating smart hinges with internal strain sensors in one step, demonstrating the versatility of DIW/e-3DP. The findings from this work provide a foundational and reliable technical solution to efficiently fabricate functional, self-monitoring, smart devices from SMPs for diverse applications.