I tested several configurations, disabling various sound inputs and outputs on my computer.
Nothing changed until I went back to the in-app settings, and changed the audio output, which solved the issue.
For some reason, Fluxer is still using the same output as before (my headset) even though the sound is supposed to come out of the screen, even after restarting the software.
This solved the echo, but not the chopping of the voice, which I resolved by disabling echo cancellation as mentioned by the author of this issue.
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Comment by @FullmealAlchemist