First tube integrated amp

You might want check out Grant Fidelity. They have some Shuguang Treasure amps that look nice.

I decided to go with separates, but I checked with some professional reviewers and all like the Primaluna Prologue 2. I think they are on sale at Upscale Audio. With your budget, you could also get separates.

PMB

I use and really like Audio Space amplifiers. The AS-3i EL34 integrated lists at about $1400. There is also a larger integrated built around the KT88 tube for more money. And if you want a remote, the Galaxy 34 is a nice amp.

I have a PrimaLuna and like it very much. The Cary amps are good as well. I would not recommend the Rogue amps if you know you like the sound of the PrimaLuna's, the Rogue amps are quite a bit brighter - they are much higher powered in general.

If I was considering an integrated amplifier I would look at the Audio Research VSi60. It's about as good as it gets. If you are patient you will find one used in your price range.

You might want to take a look at the Antique Sound Labs AQ 1003. 30 watts EL-34. Nice sounding tube amp on the cheap. divertech.com/antiquesl.html

Have fun.