JJ
I'm not familiar with Audition, sorry. But I assume it runs common plug ins.
Don't get fixated on "noise reduction". As you said earlier, it can introduce nasty artifacts and ruin the recording. Better to have a little background noise. I only used it to illustrate what could be done to your sample, but in practice I'd probably have EQ'd the bottom end. If you look at the spectral analysis of a "silent" part, the bottom end looks like a "shelf" with very few peaks. For example, in Sony Paragraphic Eq you can create a Low Shelf, select a frequency range and how much gain or attenuation you want; this would reduce the level of just that annoying part of the sound. It's not quite as simple as this, but less is best as it causes less harm. It is difficult to offer a precise answer as you need to familiarise yourself with the necessary plug-ins and adjust it so it suits YOU, on your playback system. As ever, always create a back up file of the original.
If you google DX or VST plug-ins, you will find a raft of freebies. Run the files and the next time Audition loads, it should recognise them and make them available to you.