Hello Meiatflask,
I would agree with charles1dad that SET amps generally do all the things you said you want better than EL34 amps. Line Magnetic is getting amazing reviews for their amps, and there's one based on the 845 tube for sale within your budget here: http://www.usaudiomart.com/details/649217914-mint-line-magnetic-518ia-set-tube-amp-845-one-owner-gol...
However, if you really want to stay with the EL34, and get the best of the classic sound without its weaknesses, I would second whoever above recommended the Music Reference RM9. I've seen a couple listed on A'gon within your price. I used to own one so I can attest to their musicality. Nevertheless, any true triode amp will outperform it. Just my opinion, of courses, and it goes without saying that YMMV.
I would agree with charles1dad that SET amps generally do all the things you said you want better than EL34 amps. Line Magnetic is getting amazing reviews for their amps, and there's one based on the 845 tube for sale within your budget here: http://www.usaudiomart.com/details/649217914-mint-line-magnetic-518ia-set-tube-amp-845-one-owner-gol...
However, if you really want to stay with the EL34, and get the best of the classic sound without its weaknesses, I would second whoever above recommended the Music Reference RM9. I've seen a couple listed on A'gon within your price. I used to own one so I can attest to their musicality. Nevertheless, any true triode amp will outperform it. Just my opinion, of courses, and it goes without saying that YMMV.