The DAC itself should be dead quiet at full gain. The noise is coming from the source. You should match gain on all inputs so there is no clipping and all the inputs play at the same volume.
Balance DAC and phono stage outputs to match SPL?
I have a gain control on my external tube DAC, and of course on my phono stage.
I'm sure I am not under-driving my MC cart: output 0.3mv / gain 62dB, but it is quieter than my DAC at full gain.
Streaming from Qobuz as the source, should I bother trying to get similar SPL levels for the same album, digital vs. analog, by turning down gain on the DAC, or just be happy with digital requiring less from the amp?