I am a current owner of the DT880 and HD600. Former owner of the K701, K702, Q701, K612 - and also the SRH1840.
For timbre and naturalness - and my personal favourite for classical - the HD600 does tick the OP's boxes. It would be my recommendation without hesitation. If you can find one cheap enough - the SRH1840 would be my second choice (although it is brighter than the HD600).
Based on your description ......
(Very rich sound ( especially bass), NO dry sound (live acoustic), Between Bright and Dark sounding, but a bit more towards dark.)
..... if you were going AKG, I'd recommend the K612 - cheaper and very good for classical. Not as detailed as the HD600 or SRH1840 though.
As much as I love the DT880 though - for the tastes you mentioned above, and the fact that you're driving them from an STX - I think the DT880 might end up a little bright and dry for you. From an OTL tube amp, it's a different proposition.
So in short - get the HD600 - hands down the best for classical (and your stated tastes) than any of the above I mentioned.