Is it possible to buy a decent tube amplifier in the $2000-3000 range?

I went hybrid  Rogue RP-1 Pre (tube) and PS Audio S300 class D amp.  Wyred4Sound also has a dual mono Class D that will get you in at about $3K with an RP-1 

I own the Rogue Cronus Magnum II with KT-120 Tubes. For $2,495.00 you can't beat this amp. Sounds much more like a $5,000.00 to $8,000.00 amp and the KT-120's will drive many speaker varieties
 with ease.

I'll throw in another vote for PL and Rogue. I've owned and auditioned several pieces from both - they both sound great in their own right, IME PL more fleshed out in the midrange, Rogue more dynamic, wider bandwidth, better bass control. Within the Rogue line, the all-tube vs. hybrid are split similarly, with the all-tube designs having a warmer, more complex midrange, while class-D hybrids have iron-fist bass grip and bass control.

That said, an option at the top your budget would be a used pair of Rogue M-180s. I've been running these in a system for a few years and love them for their ability to naturally and effortlessly handle many styles, with relatively inefficient speakers (ProAc 2.5s) They're also fairly tunable via tube rolling, especially via the 12ax7 phase inverter.  

I’m beginning to think I need about 200W per channel, as my 160W monoblocks are not quite doing the job.
Maybe the way to go is solid state at the price range I’m looking at ($2000-3000).
Any opinions on which way to go?

There is a Primaluna Dialogue Premium Integrated in the classifieds right now for $2700.  That would be a great tube preamp for you to consider.