2 possible thoughts……
the final sequence you are exporting….. is it a “nested” sequence? if you have a nested sequence in your final sequence, try just copying and pasting all the original, multilayer sequences in a final timeline, re-render, then export.
Another thing that used to happen to me (before I downloaded updates) is after I would render, my audio would jump around and/or go out of synch. Before I downloaded needed updates, I just put tons of razor marks and slices in my audio, somehow, this fixed the final rendered mov.
one other thought is that if the video you are working with might be set at 30fps, and your export settings might be at 29.97 fps, make sure you are always working in 29.97. If you are exporting from Affter Effects and bringing it in to FCP, make sure you record in 29.97fps.
I hope this helps, I know what its like to have a deadline and unexplainable things happen at the last second…..