Color-center defects in silicon carbide promise opto-electronic quantum applications in several fields, such as computing, sensing, and communication. In order to scale down and combine these functionalities with the existing silicon device platforms, it is crucial to consider SiC integrated optics. In recent years, many examples of SiC photonic platforms have been shown, like photonic crystal cavities, film-on-insulator waveguides, and micro-ring resonators. However, all these examples rely on separating thin films of SiC from substrate wafers. This introduces significant surface roughness, strain, and defects in the material, which greatly affects the homogeneity of the optical properties of color centers. Here, we present and test a method for fabricating monolithic single-crystal integrated-photonic devices in SiC: tuning optical properties via charge carrier concentration. We fabricated monolithic SiC n-i-n and p-i-n junctions where the intrinsic layer acts as waveguide core, and demonstrate the waveguide functionality for these samples. The propagation losses are below 14 dB/cm. These waveguide types allow for addressing color centers over a broad wavelength range with low strain-induced inhomogeneity of the optical-transition frequencies. Furthermore, we expect that our findings open the road to fabricating waveguides and devices based on p-i-n junctions, which will allow for integrated electrostatic and radio frequency control together with high-intensity optical control of defects in silicon carbide.