We have performed Raman measurements at 8 K on the $6H$ polytype of SiC nominally doped with 4\ifmmode\times\else\texttimes\fi{}${10}^{18}$ and 6\ifmmode\times\else\texttimes\fi{}${10}^{19}$ nitrogen donors per ${\mathrm{cm}}^{3}$. The 4\ifmmode\times\else\texttimes\fi{}${10}^{18}$ samples showed electronic transitions of ${E}_{2}$ symmetry at 13.0, 60.3, and 62.6 meV that we interpret as $1s({A}_{1})$ to $1s(E)$ valleyorbit transitions at the three inequivalent donor sites having symmetries that are, respectively, hexagonal, cubic, and cubic. To check the assignment of the electronic transitions to the inequivalent sites we studied the Raman spectrum of the $15R$ polytype of SiC doped with nitrogen. This polytype has five inequivalent sites: two hexagonal and three cubic. We observed two transitions at roughly 7.7 and 11.6 meV, which we assign to the hexagonal sites, and two levels at 50.6 and 54.9 meV and possibly a level at 46.0 meV, which we assign to the cubic sites. These large site-dependent differences in energy within the donor ground state point to a breakdown of effective-mass theory. The ${E}_{2}$ symmetry of these transitions in the $6H$ polytype was used to show that the conduction-band minima must lie along the line $\mathrm{ML}$ at the edge of the Brillouin zone. An interference was observed between the 13-meV electronic transition and an ${E}_{2}$ phonon of the pure crystal ($6H$) at 18.6 meV. We have proposed a phenomenological theory and obtained the electron-phonon coupling constant from a fit to theory. The 6\ifmmode\times\else\texttimes\fi{}${10}^{19}$ sample revealed a Raman continuum with ${E}_{2}$ symmetry extending to about 65 meV that interfered strongly with the 18.5-meV phonon and also with a 33.1-meV ${E}_{2}$ phonon. The ${E}_{2}$ symmetry and the interference effect suggest that the continuum retains some properties of localized levels even at concentrations greater than 2\ifmmode\times\else\texttimes\fi{}${10}^{19}$, where the metallic transition occurs. This sample also exhibited LO-phonon-plasmon coupling. The high-frequency mode (${L}_{+}$) was observed and was highly asymmetric with ${\ensuremath{\omega}}_{P}\ensuremath{\tau}\ensuremath{\sim}\frac{1}{3}$. These parameters were found by assuming the line shape to be proportional to $\mathrm{Im}{\ensuremath{\epsilon}}^{\ensuremath{-}1}$, where $\ensuremath{\epsilon}$ is the total dielectric constant. Due to the large damping the low-frequency mode (${L}_{\ensuremath{-}}$) was broadened and very small compared to the high-frequency mode. Finally an ${E}_{2}$ vibrational mode in the gap region of the SiC phonon spectrum was observed at 78.8 and 79.6 meV for nitrogen concentrations of 4\ifmmode\times\else\texttimes\fi{}${10}^{18}$ and 6\ifmmode\times\else\texttimes\fi{}${10}^{19}$, respectively, in the $6H$ polytype and at 78.6 meV in the $15R$ polytype.