Moving from integrated to separates

You will not necessarily gain much by going to tubed separates.  Many tube integrated amps are essentially a basic tube amp with just an added volume control, source selector and additional input jacks. 
 

You should not assume that you need more power or that more power is the best approach to improving sound.  You should explore all options.  More power can come at the cost of sound quality.  I personally prefer low-powered tube amps over high powered tube amps that parallel multiple tubes to achieve higher power or use tube types that I don’t particularly like.  
 

As for a specific suggestion, I would recommend the Synthesis Audio A100 integrated.  It has a surprisingly good built- in DAC.  Even though you don’t need a DAC, the amp is still quite the bargain at its price.  It uses a quad of KT 66 tubes for it’s 100 wpc rated output.  It’s baby brother, the A40 uses two KT 66 tubes per channel and it also be adequate power for your purposes.  The amps are dynamic, harmonically full and rich, and deliver a large-scale sound.  They don’t have the hard, brittle sound and upper mid-range glare I hear with some high-powered tube amps.  

The local dealer in my area, Deja Vu Audio, sells only tube amplification.  Their primary line of electronics are Audio Note, Synthesis and Conrad Johnson.  I really like the Audio Note line up, but, their amps are aimed at high efficiency speakers.  The Synthesis line spans both medium and higher wattage tube offerings.  As I mentioned above, I really like their A40 and A100 amps (both well within your price range).  For the various speakers I've heard in comparison, I liked the Synthesis amps more than I do the Conrad Johnson, and the Conrad Johnson 150 watt amp is much more expensive.  Of course it may be a matter of system matching, but, even with somewhat difficult to drive speakers, like some Harbeth models, the Synthesis amps were considerably better to me--more alive and having more "weight" to the sound. 

I know output transformerless amps are a bit harder to match to speakers (they prefer high impedance speakers), but, it would be worthwhile trying to audition Atmasphere amps.  These amps are very dynamic sounding and exciting.   

I don’t know your situation as to ability to actually hear different amps.  If you can, hear something that runs something other than KT 88, KT 120, or KT 150s.  Those tubes are almost ubiquitous where power is the big selling point, but, there are other tube types worth hearing that tend to be not as ‘hard” sounding or don’t have as much of what I would call “glare.”  Perhaps even a 211 single -ended amp from a reputable builder like Airtight (Japanese) or Wyetech (Canadian) will work.  

I have not heard any Qualiton amp so I cannot say how it compares to the Synthesis.  These days, certain types of tubes are overlooked by the market because, within a given basic design, transformer requirement, etc., these tube types do not deliver as much power, and power sells. 

But, many of these overlooked types deliver great sound.  I am particularly a fan of 6L6 tubes, but, the current market is not interested because, for the same manufacturing cost, one can make a KT 120 or KT 150 amp which will smoke the 6L6, in terms of output.  It may well be the case that you need the extra power of these tube types, but, it is still worth your effort to hear a really good 6L6 amp, or its cousin, the KT 66.  There is something quite special about some of the low-powered pentode/tetrode tubes.

I happen to own a pair of quite efficient speakers, so I have the opportunity to run some really low-powered tubes in pushpull that sound fantastic.  I run the Western Electric 349 output tube which is a relative of the 6L6, but it delivers much less power.  My amp supposedly puts out about 5.5 watts per channel.  I've been running the same tubes for well over ten years and the tubes were of who-knows-what-age when I got the amp (old stock tubes).  The amp pushes the tubes very gently.  This is a good thing because the 349s cost a fortune if you can even find them for sale.

I would be helpful to know how close you now come to running out of clean power.  With tube amps, you don't get the shrill, harsh output from the tweeter that you get when solid state is running out of gas--you tend to first experience compression (you turn up the volume but it doesn't get much louder), followed by a muddled sound lacking in clarity and imaging starts to break down.  If you hear that at very high volume levels, you have to decide how important reaching such levels really is to your overall listening enjoyment (I don't like to listen at high levels, so I don't give high volume capability much priority, but, you may differ in your needs).  You could also get a cheap meter that measures how many watts are being delivered to your speakers.  At a fairly high volume level, you generally want to stay below 1/4 of rated power of the amp, except for the very occasional peaks.

Most people grossly over estimate how much power they really need and end up chasing the wrong thing in their quest for better sound.  Nelson Pass created a company dedicated to making really good solid state amps optimized for delivering great sound at lower power levels--the company name "First Watt" reflects its philosophy.  As that company put it--"Who cares what an amp sounds like at 500 watts if it sounds like crap at one watt?"  I've heard a couple of these First Watt amps and they are among the best solid state amps I've heard.  I think the same can be said of tube amps.  Most of the very best I've heard were low powered.  Unfortunately, low-powered tube means REALLY low power.  I own three tube amps, and the one with the highest rated power is the Audio Note Kageki at, I believe 6.5 watts per channel.  My current favorite amp that I own is a 5.5 watt per channel pushpull 349 amp.  My all time favorite amp is a custom built OTL amp that I think is rated at something like 20 watts (maybe more) that a friend owns.