An R2R DAC might be what you’re looking for. Denafrips, Lampizator, Schitt, and Holo Audio all have DACs well within your price range new. I have a Denafrips Pontus II with a Hermes DDC and I like it very much and would likely sound much like what you’ve described you want. Oddly enough, the song that just queued up as I’m writing this is Warm Sound by Zero 7. I wouldn’t necessarily describe the sound of my DAC/DDC as warm, but certainly full/rich/smooth. The Denafrips Terminator line would get you where you want I think.