Wait, you've got a $200 headphone budget and $300-$400 amp budget?!? Sounds like you've got it backwards there. Get the best headphones you can afford and an amp that is adequate to power them.
Hmm, what would I do with that much, getting the best for jazz and cymbals... Probably a used HE-5 for $400-ish, and the remainder on an amp.
Something exactly like...
this. You don't even have to pay for separate shipping that way...
But honestly, for $600, you'd get a better rig for jazz and cymbals by getting a pair of speakers with planar magnetic or ribbon drivers. Along the lines of Magnepans, Infinities, Apogee, Carver, or the like. Set aside $100 for an amp and spend the rest on the used speakers (off Craigslist) that suit your fancy. Alternatively, save up for new Magnepan MMGs ($600) and an old NAD integrated amp ($100-ish) or Adcom power amp ($150-ish).
But that's my speaker bias shining through. You may not find them practical given your situation (right now for me, all but a pair of bookshelf speakers are in storage), but I certainly find them to sound far better. Done properly, you'll get that "in the club" feeling.
That said, my HD 600s and uDAC sound great with jazz (John Coltrane, Miles Davis, Louis Prima, etc.).