One way will be to use the playhead to make a selection on the timeline, right click and select 'Produce Timeline Selection as' choose the format you wish to use. If you have other media on tracks in the selection, you can disable them so they are not included.
You can right click the clip with video and audio to separate audio and video, then disable the video track so that it's not included.
There's also an option to Export Audio only under the Share menu.
I have had success with exporting each individual audio track, one at a time, whilst the rest are turned off (set to "disabled"). The audio export takes much less time than a video export.
Also, be sure and export as WAV, to retain quality.
Be sure and export the entire timeline per track, so that you can drop each file into the beginning of another program/back into Camtasia placed at the beginning of the timeline, so all tracks are in sync.