To answer your question directly: yes, stretching UV islands disproportionately is unavoidable when switching a texture from square to rectangular.
This happens because regardless of its aspect ratio, a texture always occupies coordinates from 0 to 1. When you select a rectangular texture, you don’t change the actual coordinates, but rather how they are displayed in the UV Editor. The main UV tile still ranges from 0 to 1 on both axes, even though it now looks rectangular. This is exactly why your islands stretched — they simply preserved their original coordinates.
To fix this, you need to scale the UV islands back in the opposite direction by an amount based on your texture's aspect ratio.
To avoid this issue in the future, simply assign the rectangular texture in the UV Editor before unwrapping or creating your islands. This way, the tools will already recognize the rectangular proportions and calculate everything correctly from the start.