Your DAC is good for about 7 mA which isn't too bad. 1V RMS into 10K really isn't too bad.
The problem is certain OP-AMPs "Sound better" when used for Audio. They can generally be found in the selection guides on the manufacturer's websites. An old OP-amp was the NE5535, but there are better ones that that now.
Here is an article to start:
**broken link removed**
Since your starting from the bottom, there are components that are "sprinkled" on a circuit that are not usually shown on schematics. These include power supply bypass capacitors. These are low values capacitors located really close to the OP-AMP power pins. The higher the bandwidth of the part, the easier it is to oscillate.
It's known that Audiophiles will go and replace OP-amps for better sound.
Here is an example selection guide:
https://www.electro-tech-online.com/custompdfs/2012/09/slyy013g.pdf