What is a very nice sounding tube preamp?

I have not heard them, but have read good things about Rogue and Doge tube preamps.

There is a VAC Standard preamp with some upgrades for sale now in the classifieds for about half of the amount you mention in your post that would seem to me to meet your criteria. While I have not heard this particular model (I have heard their more expensive offerings), I have found VAC to be among the handful of manufacturers who make unfailingly musical products. I can also vouch for Rogue, another manufacturer of quality tube equipment.

Joule Electra LA 100 Mk III is a good one, and you should be able to find one at a fair price. Read the reviews from when it came out. I have one, and like it very much.

I own a Vac Standard and have to say it is a very good preamp.
I have tens of thousands of equipment. And the VAC Standard is a match.
I have mine AC from a Furman REF20i via a Pangea AC9SE powercord.
I paid $1,400 for mine here. With the upgrade that one has YES grab it.
The VAC Standard has a separate power supply, and is RCA in out only. four tubes two 12Au7 and two 12AX7. Runs very cool. I leave mine on 24/7 thhree years now.
It really IS a $4,000 preamp.
That is only the second one I have seen. I bought the first. So they are rare enough not to pass up.

In addition to the previous recommendations, all very sound,

I own a new , Jolida Fusion, owned a Audible Illusion 3a bought used. Both at your price point and very good. You might also consider Quicksilver.

Enjoy and let us know your choice.