First, try trashing your FCE preferences. Trashing your preferences is usually the first step to take when Final Cut acts strangely.
You can download Preference Manager (free) from Digital Rebellion to have it done for you:
If that does not work, you can also try copying the entire card to a hard disk drive, and attempt to Log and Transfer from the copy.
If neither of those suggestions work, post back.
Log and Transfer does not work with Standard Definition Media, and there are anecdotal reports the Log and Transfer gets quirky with cards that contain mixed media - HD, SD and/or still images.
Also, in the log and transfer window, click on the gear in the top center portion of the screen, and from the drop down menu, choose preferences:
When the preferences open, make sure the AVCHD import audio is set to Plain Stereo
Does that help?
If you have made a copy of the card as a back up, does your camera have a feature that will rebuild the card database (as it possible the card directory has become corrupted)? You may want to try a rebuild the directory, but make sure you have a back up copy of the card first.