Suggested upgrade paths

Is it your system or your brain & ears?  I hope it is a system component problem.

I have relatively mild tinnitus (I am 81 years old and a couple of years ago I was exposed to a lunatic DJ at a wedding, I had my -30dB ear plugs in but the spl was insane - the plates on the table bounced - I left, unfed, but the damage was done).

As a result of this the brain can develop a condition, hyperacusis, where it overcompensates for loss in the upper treble and overreacts, making certain frequencies, above a certain volume, become shrill, harsh and even painful.

This has happened to me and has changed my listening habits. Certain high treble frequencies, above 60dB or so, fire up the harshness.  All sound sources do this so it is not an artifact of my "entry level very high end" system (Esoteric K-01XDSE, Rockport Atria II etc.).

I am affected mainly by orchestral crescendos where the violins are going fff, Violins have a very large harmonic component to their sound, but the effect is still there with an a7 sin wave at 3520 Hz.

I wish you good fortune in the search for an answer whatever the cause.

I have owned the last series Lintons & I do not suspect that the tweeters are your problem. Unless there is a mechanical issue with them which would be rare for both speakers to have the problem..unless you damaged them with loud volume. 

I would start at the source. Do you use any PEQ or EQ on your Eversolo? Any boosted frequencies? How do you have the voltage output set up? Fixed output or Variable Gain (volume control)?

If you use any extra features on the streamer turn them off. Set the output to fixed. If still a shrill volume problem set the output on the streamer to volume control & set it around 75% of full volume. You want to rule out that maybe your streamer is sending a distorted or clipped signal to the dac. Or maybe the streamers output is too strong/hot & it's clipping the dac. I have encountered both of these problems in the past. Some of these devices have very poor DSP implementation...adding distortion that outweighs any tonal correction they may impart. 

Your integrated has plenty of power & low enough distortion that it should not be the culprit. The internal dac is no prize winner but it as well should not be causing any issues from a modest streamer like the A6. But if I was wanting to upgrade then the Dac is where I would start if digital streaming is my go-to source. You could go all out with a Venus 15th or a notch lower at the Pontus 15th & a new streamer. 

I am now a strong believer in the R2R topology. I find them to take a bigger step away from digital artifacts than the chip based dacs I've personally used. Laiv audio makes some quality R2R dacs as well. Halo audio is another popular choice. 

It would be nice to figure out your current issues first though. I just suspect some form of distortion or noise getting into what should be a clean enough signal from the components you have. Just for giggles throw a large sheet of tin foil over your streamer..your device may be picking some noise up that when it is really amplified you are hearing it. I can think of a few times I've covered my tube equipment with tin foil making me realize it's too close to a router or transformer component. 

I had issues with some type of distortion a few years ago & couldn't figure it out. I finally figured it out by covering this Elac class D amp I was using at the time. The Elac DPA-2. Covering it with tin foil made the light fuzz go away. Looking for answers I had found a review on the unit where the guy mentioned he had an issue with the amp picking up noise from a FM radio or something. I would have my TV on with the volume off lots of times when listening to music. This amp was picking up hash from the TV. Crazy. Tin foil is my friend though. 

The Eversolo DMP A8 has a pretty good DAC, and streamer.  You could try one of those for an in-home audition.  This DAC is completely different from the A6's.

I have an A8 in my 2nd system with an Elekit 8600 and Harbeth P3ESR speakers and it is rather nice.

There is also an even better A10.

If you bought the A8 you could be out-of-pocket only about $1,500 as there is a market for a used A6s at about $500.

OP

@nucleardog61 

When you wrote that as the volume increases, the Treble starts to sound shrill. I have encountered this, and the culprit is that the AMP cannot give the power to the speakers. when we replace the amp, problem solved. I have owned a couple of Rotel Power amps, and they do sound good, but they all seemed to run out of steam as the volume comes up. even driving horn speakers

Being cost sensitive, I suggest used. a Used Pass or Luxman may be the fix. 

I have seen good reviews of the Litton so that is ok for now. I truly suggest a powered sub like the RELs and SVS line. BUT I truly believe that the possible culprit is that your amp is running out of steam as you punch in the power