Are there any tube amps that don't give off any---

Yes, the Purifi module is class D and I agree that class D amps can sound too sterile. I had an NCore based class-D amp previously and that is how I would describe it.

However, the sound signature of amps based on the newer modules (particularly the Purifi) are going to depend more on the input driver design than the class D module. The Purifi module has vanishingly low noise and distortion, but limited gain (as well as low input impedance), so a front-end driver is required in most situations.

I believe (although I haven't personally heard) that the right input driver can create a sonic signature that is pretty similar to whatever you're looking for. I could be wrong, but I'm giving this a try myself by building a set of DIY monoblocks using the Purifi module so that I can experiment with different buffer/drivers. I'm planning to build (or buy) a few different options to see what works best in my system, including building a triode-based buffer. 

The VTV amp I linked to above uses a triode buffer that looks interesting, but I think I'm going to design and build my own. 

I think VTV has a pretty liberal return policy, so if it seemed like it might work for you, you could give it a try.

+1 LTA Berning designed OTL.
I have the MicroZOTL headphone amp.
SS detail with tube musicality and not hot.

Just dawned on my that you are the Fender Guy...

Good luck with your search.

DeKay

Like others have said, Carver. I own one and you can touch the tubes it’s so cool. My new MacIntosh MAC 275 is similar it runs very cool. Make sure that you get generation six

Try a hybrid.  Tube front end but solid state power amp.  Small preamp tubes don't produce much heat and I find the combination very pleasing and less expensive.