Switching adapters are actually not that bad. Many "audiophiles" swear by toroidal transformers, but that's just a myth along with a lot of other myths, although they have some benefits, but then switching transformers have just as many benefits over toroidal trannies.
I would try with a ferrite ring. Buy a ferrite "clip-on" sleeve (costs about 0.5 USD) with an inner diameter that's 4 times bigger than the outer diameter of the cable between the wall-wart and the DAC. Make two loops in that cable, then clip on the ferrite ring (about 2 inches from the DAC connector) so that the cable passed through 3 times and around it two times. It will solve a lot of the noise issues you have.
Ferrite rings can be used for a lot of noise problems, and you can't oversize, the bigger the ferrite ring and the more passes, the better the performance, but usually, the aforementioned solution suffices.
Another really important thing about power supplies is the grounding. There are two common issues that pop up with grounding in audio applications, of which the main one is the ground loop. The ground is what the word says, it's a connection to the earth, and it's universal, that means that the ground (the black wire) of your audio path connection is linked to the ground of your electrical connection. Electricity - like any energy - will always follow the easiest path, aka the path with the least resistance or impedance. The audio signal is attenuated by lowering the resistance to ground. that means that what separates the audio path from the power supply ground, is maybe a single 100kOhm resistor or a single 100kOhm potentiometer (variable resistor). That means that noise and EMI on the ground connection shared by all the devices (which also contains noise generated by those devices, in the case of digital devices, that's a lot of noise), will sneak into the audio path and feedback on itself. There is an easy fix for this, and that is decoupling the power-side ground connection from the audio circuit. That will not cause the audio circuit to lose power, because it's always connected through the audio-side ground connection to ground. A ground loop has a distinct character, it's a kind of lower pitched hum or rumble sound, not just hissing noise or the higher pitched scratchy EMI noise. So if you hear a ground loop, just cut a small piece of electrical tape, and tape off the negative side of the DC connector to your DAC, and it will go away.
Since it's 12V you need, another cheap solution is to buy a Roland 12V filtered power supply. Those are not expensive, but they just work. They have a small enclosure on the DC cable that contains a combination of magnetic filtering with a ferrite core and a rudimentary LC network that filters specific frequency ranges that have no place in the audio signal. Another solution, if you're a guitarist, is to get a pedal board power supply. Those mostly use toroidal trannies (one per output, they have multiple outputs), which makes them more expensive, but they also use a voltage regulator per output and the outputs are insulated as far as that is possible from each other, which eliminates a lot of noise. There is a limit to what can be done in terms of making a mains connected power supply noise free, it's never as clean as battery power, but in comparison to plain unfiltered wall warts, there is a huge difference. If you're not a guitarist, investing in such a power supply is just stupid.
If you want to have the cleanest possible result, you can buy 10 NiMH "baby" cells (which usually put out 1.25 to 1.3 V DC), and roll up two transparency sheets to line them up in series. All together in series, they will put out 14.4 V at maximum load and 11.8 V at minimum load, which is the voltage range commonly referred to as "12V", and a perfectly safe range to operate most 12V rated devices with. At an averag 4Ah of energy load for baby cells, that DIY battery pack will last you a very long time between recharges, and NiMH batteries will recharge rapidly, so that's a pretty good solution also. That will give you perfect noise free DC power.
The problem with DACs is that they contain both a digital part and an analogue part. Digital parts cause noise, that's just what they do, so sometimes it pays off to move around the parts on the board to find a placement that leaks less noise. It's the same with trannies, valves, etc... there is always a placement solution that leaks minimal noise.