We study the self-frequency shift of continuously pumped Kerr solitons in AlN-on-sapphire microcavities with Raman gain bandwidths narrower than the cavity free-spectral range. Solitons are generated in ∼230GHz microcavities via high-order mode dispersion engineering. The dependence of the self-frequency shift on soliton pulse width is measured and differs from amorphous material microcavities. Our measurement and simulation reveal the impact of frequency detuning between the cavity resonances and Raman gain peaks, as well as the importance of all three Raman gain peaks. The interplay between the Raman effect and dispersive wave recoil and a potential quiet point are also observed.