If considered in units proportional to energy, such as inverse/cm, the Raman shift does not depend on the energy of light used except that the photon must have more energy than the transition being stimulated. Naturally, in units of wavelength, which is inverse to energy, the size of the shifts will seem different at different wavelengths, and that is the best reason spectroscopists work in inverse cm, not wavelength.
The lack of dependence on photon energy can be found in pretty much any description of Raman scattering, for example the Wikipedia article:
https://en.m.wikipedia.org/wiki/Raman_scattering
The photon interacting with the material generates or absorbs a phonon. The phonon energies are determined by the structure of the material. To conserve energy the change in energy of the photon must equal the change in energy of the phonon. So the shift in energy of the photon is the same regardless of the photon energy
In addition to Albert's answer, I could add that if you see an apparent change in Raman shift with wavelength, please check if the peak position is approximately constant. This may suggest it could be a contribution from fluorescence. Alternatively, if the peak position and the Raman shift changes, it may be a new peak arising from resonance Raman scattering, if your laser wavelength is resonant with electronic transitions in your sample.
By using different wavelengths you might see very different things, as you excite different vibrational modes of different components in your sample:
Article Excitation wavelength-dependent changes in Raman spectra of ...
But also check this thread:
https://www.researchgate.net/post/Why-does-my-SERS-Raman-spectrum-peak-positions-change-significantly-when-switching-from-785nm-to-637nm-excitation-wavelength