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

I think what makes your situation a little different is you’re not just looking for improvements but also different sound characteristics.  Because of this I think it’s possible for you to make significant strides toward the sound you’re looking for without necessarily having to spend $6k on a preamp (I’d refer back to my recommendations above that don’t cost anywhere near that much but would sound much more like what you’re looking for).  
As for the preamp, I and many here have found the preamp to be as or even more critical than amps, and this can be even more pronounced with speakers that are easy to drive like yours are.  You’ll definitely want to upgrade your streamer as well, but a stopgap might be to get an upgraded linear power supply for the Node where a good one from Teddy Pardo, SBooster, or LHY costs around $400 and should at least hold you over until you can make a meaningful upgrade there.  That’s probably what I’d do FWIW.  Anyway, just some further thoughts I hope are at least somewhat helpful. 

I agree with Freddy that an AN Oto is a good choice, but you will have to look for something used.  While I don’t think that the Cobra is a bad amp, it is not really an AN amp and it is not nearly as good as various Synthesis amps, like a DC 96.  I heard a comparison of the Cobra with the comparable Synthesis amps and much preferred the Synthesis.  I believe the speakers used in the comparison were AN-J’s.

Yes, preamp first. As it is known... the heart and soul of your system. I would definitely recommend a used tube preamp. This really sets the stage for the rest of your system. I'd look for a used Audio Research LS28... or go back just enough to capture one in your budget... LS27... LS26SSE. They go LS## then LS##SE. Conrad Johnson and VAC are also classical musical, natural and detailed sounding preamps. My objective is to have my system sound like music... not like hyper detailed sound with slam... which does not communicate the emotional connection. Many folk go for incredible detail and slam... but loose the music along the way. 

Yes, I would start with the preamp. I am not suggesting that you spend your entire $6k budget on that component, unless it makes sense. It might make sense if you expect to have more budget in the foreseeable future, if not, you may have to scale your purchase. My point was that you should make the changes one step at a time, so that you can get a feel for the impacts of each component as you add it to the system. Keep in mind that the first upscale preamp you buy may not be your last, and you may have to try a few to get one that best fits your system. I am on my fifth, but pretty satisfied. Best to go used for the sake of economy.

A lot of recommendations for tube preamps, and I don’t disagree with that path. However, make sure it fits your lifestyle. My gear is on all the time and sometimes plays all day. For that reason, I have not ventured into tubes, though I did have a Schiit Freya that gave me the option use them.

Be sure to allow some budget for decent interconnects, they will make a difference as you get into better gear. Acoustic Zen and the newcomer Veritas, are both reasonable and decent.

AN gear is a little different animal from most other HiFi gear.  Their integrated amps will beat the pants off most other separates you can buy for comparable prices. Start with a Level One AN integrated amp.  The Oto is wonderful if you can find one in your price range.  The Cobra is not far behind it and has a built in DAC which sounds very good btw. Most with AN speakers will tell you they sound better with AN electronics and wire anyway.  Then if you’re ready to upgrade look at Level 2 and 3 AN components.  I say if because with the Oto you might be happy for quite a while and it will give you some time to upgrade your source components.