Looking at the board... apparently a switching converter is used to create +/- 15V for the discrete opamps on the opamp board, from the +5V power supply. Can you spot the two tiny "472" inductors?
Also there is a CLC filter on +5V input, which makes the output impedance of your LM317 power supply irrelevant. No matter how low the power supply impedance is, it'll be in series with the filter anyway.
On the DAC board we have lots of LDOs, presumably turning the incoming +5V into 3.3V and the other voltage ES9038q2m requires, I think it's 1V2 but I'm not sure I remember correctly.
Note, from the amount and type of decoupling caps visible, it looks like these guys don't own a network analyzer.
It is unnecessary to make a microvolt noise power supply to feed a LDO since the LDO will have a good amount of PSRR and will add its own noise anyway.
What is important is to have low HF noise because LDOs usually have low HF PSRR. ALso low common mode noise is important if you use an AC-DC switching supply. Since you use a linear power supply, these conditions should be satisfied without problems.
NEVER TRUST AUDIOPHILE REGULATOR DESIGNS unless they come with full specs, output impedance graphs, and specs about PSRR, noise etc, and especially stability. I've tested a number of these "audiophile regulators" and... some are pretty good, some are unstable, there was even one that managed to pick up some AM radio.
All regulators are closed loop feedback systems. They output a certain amount of current depending on how far the output voltage is from the desired value. You can represent that with an error amp with voltage gain A, followed by a transconductance G. This is explicit in most LDO datasheets which show an internal schematic with an error amp and a PFET as the transconductance device.
If the output voltage changes by dv, then the output current will change by AGdv, and the output impedance of the regulator is 1/(A*G).
An important point to consider is that the impedance Zo, which is load impedance in parallel with the output caps, is part of the feedback loop gain. So, the loop gain is LG=AGZo.
To have a stable feedback loop we need LG to have less than 180° phase shift as it reaches unity gain.
This is harder to do if LG has a high value at DC, because it will have to fall down all the way to unity gain without having 180° phase shift there.
And... adding transistors, as you did, increases loop gain while also adding more poles, therefore more phase shift. This means it makes the whole system less stable. These regulators "boosted" by a transistor are known to be finicky and prone to oscillations for this reason. This is most likely your issue.
Honestly, since the board already has DC-DC converters on it, I don't see any reason to use a linear power supply. You could simply use one of these Meanwell PCB mount bricks or a high quality wall wart. You can use the medical version for lower leakage and common mode noise.