Likely cause is a broken / worn / popped out spring within the handles. You can investigate (take them off and have a look) with a view to fixing or simply buy a new set of handles and replace. You don't need a complete mechanism if by that you mean the lock etc. It's nowt to do with that
I will say the exact opposite, those handles are probably not sprung so rely on the spring in the mechanism to keep the handles horizontal, just remove the handles and leave the spindle in place and see if it's already half turned or slide it out and look at the square hole through the door to see if that's level, my guess is the lever handle is sagging with its own weight because of the slop in the mechanism spring so it may need a new mech or possibly just a gearbox if it splits from the main strip, you could fit some spring cassettes inside the handles if you can find some narrow enough, they would help return the handle back to horizontal or close to
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