Why do my ears ring with my new amplifier?

"I was grasping at straws to see if the system sounded better/worse with headphones; maybe more neutral by eliminating the room. A separate issue from the ringing I guess."

Its certainly worth a try just to see what happens, but not only would you be eliminating the room with headphones, but the amp, speakers, some of the cables, and possibly the preamp depending on where the headphone amp is.

"I don't think that the high frequencies are the problem."

You may not think HF's are the problem but I think you might be mistaken. A funny thing happened to me a while back when I listened to a DIY Class D amp. Initially, the first 5 minutes it sounded all right but certainly not perfect, transparency was amazing but.....after about 15 minutes I started getting a headache. I'm not being sarcastic, this really happened. The problem immediately went away when I replaced the amp. The REAL dilemna was I never could figure out what was causing the headache. The same amp in the owner's system didn't do this.

If you're not listening too loud, which could certainly cause ringing, you most certainly have an issue with the upper frequencies. The cause is less clear but it could include the amp/speaker interface, most likely IMHO. Room treatments and speaker placement may help ameliorate the issue. It's tough to tell without knowing everything. Do you have any audio buddies or a wife that could chime in with their opinion? It might help narrow things down a bit.

I've face this problem a few times, I think because of listening too loud, some amp is too detailed and transparent ,there are too much detail in the front row , you have to turn up the volume to try to hear the dept in the back row and also to distinguish the singer. The bass energy surge when volume is up , I've to see doctor twice, I think this amp doesn't suit you, you dial up the volume because you donot find it involving , just my thought.

It's a sign from God to buy better equipment

Its time for an upgrade.