By default, the PS Audio DirectStream DAC is a wonderful performer, but I wouldn’t characterize it as analog sounding. It can have some etchedness and brittleness to it when compared to more expensive endgame DACs, but it is quite hard to beat in the $3K used range. I owned one for many years, and I went through all the possible modification upgrades (upgraded transformers, linear power supply mods, isolation tweaks, etc) and all made a profound difference in performance that brought it closer to analog. I did find the Lampizator Baltic 3 to be more “analog sounding” but less defined and resolving as the PS Audio.
For $3K used and “analog sounding”, consider R2R DACs, including Audio Mirror Tubadour III SE or IV SE or Aqua La Voce.