Warm, rich, smooth, 3d tube integrated amp, budget $6000

Find the Canor 1.10. An incredible sounding integrated. Retails for $6900, but I bet a dealer would meet you at the price.

@twoleftears + 1  I had one of Icons Phono Section and it was quite nice. 

Icon Audio is often overlooked but they are excellent value for money.  And yes, they are imported and for sale in the US.

https://iconaudio.com/integrated-amplifiers

I am a broken record on this, but, the integrated tube amp that I am a BIG fan of is the Synthesis Audio A-40 (40-watt, pushpull, KT-66 tube).  It comes with a killer good DAC included.

For a price a bit out of your range the Synthesis A-100 (100-watt, four KT-66 per channel) is the way to go (yes, it also has a built-in DAC). 

Both amps are among the best sounding tube amps on the market.  The sound is both rich and bold sounding, without being sluggish or murky as some other warm or rich sounding tube amps can be.  

Felkis, Audio Hungary, Raven (yikes not them!) 

I have the AH Qualiton X200 and freaking love it.

Mastersound. Not sure if $6k is enough for a used Evolution 845 or Gemini but Mastersound would be my 1st choice for a reasonably priced all tube integrated amp. 

Line Magnetic would be a good pick for that sound signature depending on your power needs. Some of them, especially the high powered SET amps, generate quite a bit of heat.

Feliks Audio ?