I’m also searching for the same topic and I figured out a way to achieve it
You may choose you playback device to your computer’s sound card, in my case, it’s “speakers/headphones (Conexant Smartaudio HD)”
Then, I choose recording device to my mixer “microphone (USB2.0 Device)”
I plug my headphones to the computer to listen to the backing track while recording using the USB mixer.
The recorded new track was purely the sound from the mixer without the play back of the backing track
Hope it also works for you.
Personally, I do what Clarinettse does, below. I know what I sound like, and I can see the waves on the Audacity screen for confirmation, so I’m OK with this.
I know that if you don’t have confidence in what you sound like, this extra audio signal can be invaluable. I happened to be perusing the manual for Allen & Heath ZEDi-10FX mixer and I feel it has the controls you are looking for. The mixer is a top quality mixer and is priced accordingly.
On this mixer you can specify separate routing for the headphones and main mix out. Alternatively, you can specify whether to send separate channels or the main output to the USB bus.
Be forewarned that due to extra controls you may find this mixer a little more difficult to use than others.