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The dialogue is warning you that you are going to change your imported document from 16 bit to 8 bit (or 32 bit to 16 bit) . In most cases just carry on - or if you need the higher bit depth - click "No" then change the document you are pasting into to the higher bit depth ( Using Image - Mode) then re-try.
So I did this, but it still drastically alters the appearance of my image. Is there not a way to avoid that?
Bit depth doesn't "drastically alter appearance". Then it's something else.
Can you post a side by side screenshot?