3way replacement for my 2way

I’m thinking of replacing my Triangle Comete EZ Bookshelf speakers with some 3 way speakers and seriously considering purchasing either 2nd hand or new.

My Triangles are good about 85% of the time, they’re somewhat forward, not very forgiving and lack that warm feeling of a 3 way.

My Triangles currently are on the company’s accompanying stands about 24” from the back wall and about 20” from the side walls and towed in projecting to my couch’s center. My room is 12’x 11’ w/ a 10’ ceiling, in addition the room has a tray in the center giving the center an additional 1’ of height. The floor is carpeted, no furniture other than a couch against one wall facing the equipment and record storage racks on the other wall, one window on the side wall is treated with thick room darkening curtains, basically the room sounds pretty good.  

I’m running a Carver Crimson 275 Amp that’s hooked up to a Parasound ZPre3 Preamp. I mostly listen to vinyl on my Orbit U-Turn TurnTable equipped w/ a Ortofon Super OM20 Cartridge stylus that’s hooked up to an IFi Zen Phono Preamp. 

I get that it’s a “Mid-Fi” set up but it lacks some warmth that I feel is a result of my 2 way’s extreme forward output. 

Like I mentioned prior, my Triangles sound great at times, they just always have that in your face high going on, sometimes more than others. 

I’m looking for 3way speaker recommendations in the $2k - $3k Ballpark that have a nice balance of Highs, Mids & Bass. 

I know there’s some really great speakers out there and not to sound obnoxious, but please don’t recommend some great pair of used speakers that even if I spent six months full time searching the internet I won’t be able to locate. 

Not into Klipsch,No B&W’s or Polks. My music tastes are Blues,Jazz, Big Band, Reggae & Classic Rock if that helps. 

I’d love to hear your thoughts.

For what it’s worth, I found ATC scm 40s used for around 3500, and got a pair for a couple of hundred less that had some dings on them. I agree with an earlier poster that these are head and shoulders above the other suggestions. On the other hand, this is still above the poster’s budget, and may be irrelevant as a result.
I had small Triangles and found them too bright, I did however get good results by adding 2 (definitely not 1) subwoofer and setting them at between 80 and 120 Hz. That would leave you tons of money to spend on a better phono amp (which I see as the weak link).  The Vincent Pho 701 would temper down the Triangles whilst improving mid range and bass. 
My 2 cents:

Try making up your own ac cable using tinned copper with cotton insulation.  Duelund and NOS Western Electric are easily available.  Much more body and warmth.  Throwaway money.

Finished cable brands that come to mind are Huffman and Luna (Canada).  Using just one in a system should satisfy. 

Best wishes
I like the idea of the Lintons but I have not heard them.

I am a recent fan of Focal.  I would look at the Focal Chora 826.
If you listen to just music and that’s all I would look into pro speakers and pa amps, for music that configuration can’t be beat by home hifi even at the same price points. Now if you watch shows and movies also you have to think hard what you want more near silent operation or some fan noise with pro equipment, I’ll say this listening at a moderate level of volume the fan noise you do not hear at all, I have crown amps xls 1502 x 3, xls 2502 x 2 and an xti 6002 paired to two Cerwin Vega xls-215 and the 1502 are paired to a klipsch 250 c and Bose 301 running in parallel so the bose are running in 4 ohms and the Cerwin Vega xls-215 are 6 ohms and I believe the klipsch 250c is 8 ohms, and my subs are a Cerwin Vega cvx-21s and Cerwin Vega el-36dp which out of my whole system the pro subs sound the best paired with music so like I said if you listen to just music look into pro and forget about that home hifi crap scams