I have no idea even what "synch lock" does and never touch it.
I do a lot of J and L cuts though.
Try with synch lock off.
Are your audio tracks checker boarded ( A/B'd) ?
I usually alt-drag for J and L cuts
With the sync locks off, the video clip will move but the audio won't follow it at all. The J and L cuts work when I alt-drag with the mouse, but I can't fine-tune them in trim mode like I've been able to in other software because the audio track has attached itself to the wrong clip.
Is it really attached, or just "snapped"?
How does it attach if its on another audio track / layer?
Instead of the Trim mode, just use the Rolling Edit Tool (N) with ALT held down.