Just throwing a couple of ideas...
Try cutting the whole thing in 29.97 not 23.976
23.976 translates onto 29.97 using a pull-down but not sure it works the other way around so you should work to the higher frame rate.
Otherwise your 29.97 will only play 23.976fps of the 29.97 frames e.g.: skipping or frame blending to try to match second-for-second.
When you sync the dailies you might also want to create new master merged clips - link the audio and video in the timeline (cmd-L you should see the clip names underlined) then drag the clip to the browser creating a new master clip with both sources.
Try using those new master clips on your edit see if you still get sync issues.
I'm guessing if your final output is 24p then its not the frames per second that will be giving you the "different look" - more that you shot interlaced video and it looks "cheap" versus the 23.976?
Might be worth conforming or standards-converting the 29.97 rushes in that case and cutting on a 23.98 timeline.
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