$6k for DAC/Pre/Power Amp for Audio Note ANE/LX

In your price range, it makes sense to consider an integrated amp, particularly if you go with tubes.  Many tube integrated amps are essentially the same as a tube power amp with just impute jacks, an input selector knob and volume control added to the power amp--these are cheap additions compared to another chassis, power supply and input tube circuit.  

Of course you should consider used Audio Note amps, such as the Oto, which sounds very good with your AN-E.

My favorite brand of tube gear is Synthesis Audio (Italian).  I've heard several of their amps with Audio Note speakers and the combination is VERY good sounding.  One of my favorite integrated amp of theirs is the A40 (40 watts per channel) that comes with a quite good DAC built in.  The combination is around $2,000 more than your budget, but, its performance is WAY better than one would expect at its price.  I could easily live with the A40 even though I love my current amp and linestage whose price is now probably north of $80k.

Given the quality of your speakers. I would consider investing in one or two components at a time and raising your investment level. I would recommend either spending your current budget on a preamp or on an integrated amp. You will get a big improvement in sound quality. Then after a year or two, consider purchasing an amp (if you had purchased just a preamp), or a high quality DAC.... and finally the year after a high quality streamer. 

My rule of thumb is that you want to invest in your components about 15 - 20% of the value of your whole system. So speakers around 30% and each component about 15%... preamp, amp, DAC, and streamer. 

Long term it is cheaper to do one component at a time as opposed to small increments of increase to all components at once. You still enjoy each upgrade, but when you are done you have a synergistic fully realized system. You can concentrate on each component exclusively and carefully choose your tuning of the sound as you go. 

Anyway, consider this classic audiophile process to upgrade. 

Agree with @ghdprentice that it can be tough to find three pieces at once that synergize to your liking.  I’d be more inclined to get one component and then see where you wanna go from there.  That said, I think pairing this pre and amp would be a sublime combo given what you’re looking for.  Add a Denafrips Pontus 15th and you’d be right around your budget, and you could add an Iris or Hermes DDC (for i2S) later for a significant performance boost for not all that much $$$.  Just one idea FWIW.

https://www.usaudiomart.com/details/650228633-don-sachs-model-3/

https://tmraudio.com/fresh-arrivals/valvet-e3-stereo-power-amplifier-1-3/

A contrary view but I would first purchase a new DAC before the pre and amp.  I have a node 2i that I use with a vintage but updated Museatex Meitner DAC.

When sent away my DAC to have it "refreshed/modded" I did not stream whatsover as without it in my system I thought the DAC in the 2i was lacking in all the ways that you describe your sound.  Placed the modded DAC back in the system and all returned to sounding great.

I know you said you’d prefer separates but you really need to consider the AN Cobra.  I listened to it with your speakers and it sounds VERY nice.  Used it fits the budget too!