First rule of FCP… don’t mix frame rates and, don’t mix Video speed (23.97 fps) with film Speed (24 fps ) material in one time line.
FCP “frame rate” is the video frame rate setup in the sequence setup, in your case 23.97 ( Video speed) . Audio “frame rate” as you suggest does not have anything to do with the timeline frame rates.
When you import video and audio using a camera and FCP’s log and capture, you are capturing at the video frame rate. If you are using a dual system, you capture at the video frame rate matching the video rate shot. ( again, in your case 23.97 Video speed) Make sure this is all set up in the session setup/ capture setup and sequence setup.
When you import audio standalone due to the use of a dual system, and drop the audio into the sequence, there is a rate conversion done by FCP to match the current timeline, even when the audio does not match the video frame rate. Therefore the audio may be slowed down or sped upa s the case may be. One of your issues is that FCP sets an NTSC VIDEO flag on the audio tracks even though they are shot at 24 ( film speed ).
You are probably seeing the video and audio in sync at the head of each clip, and as the timeline is played the audio drifts. you can ( although not recommend) chage the audio speed to match the video. I believe it’s 99.9% or 100.1% depending on your drift problem. This will cause the audio to be resampled. ( for us audio purists, this is scary, as the audio has now be re-sampled twice)
Here’s another way… ( can’t absolutely promise success )
1. Select the problem sequence in the FCP browser, right-click it and choose Export > XML.
2. Open that XML file in a text editor.
3. Look for the tags <ntsc>.
If they all say <ntsc>TRUE</ntsc> or if they all say <ntsc>FALSE</ntsc>
then you don’t have a problem.
But if there is a mix of TRUE and FALSE values, you need to change them so they are all the same. match the option to your video clip settings.
When you add a video clip to a sequence, FCP knows what frame rate to use. It just looks at the video clip properties. Audio doesn’t have a frame rate like video, but FCP needs to assign it one because many editing operations are tied to the video frame rate. So when you add an audio clip to a sequence, FCP has to pick a frame rate for it. That choice is apparently made based on some combination of sequence and capture presets, and may depend more on what settings are in the cache than what are currently selected. And while FCP seems to be able to adapt to whatever frames-per-second are appropriate, it needs some help to get the NTSC values right.
hope this helps.