If you're hearing the sound as "sterile", it's likely not the HD-600. The midrange is where these cans are at their best. They will also reveal what's going on with upstream equipment. I've only tried a lower end Headroom product (old style Supreme), but was very unimpressed with the sound.
The Clou cable improves detail, bass extension, definition, clarity...it's a definite upgrade in interconnects and raises the Senns to a higher level of performance. However, I'm not sure that addresses the sound characteristics you're looking for.
If you can find one, try out a TDS passive audiophile. This is a processor that I posted on in the tweaks forum a while ago. I've used it with AKG cans to bring out the midrange and achieve a better tonal balance. With Senns, I don't use it, as the midrange becomes dominant, making the highs and lows seem recessed.
The more I look at what you wrote that you're looking for: "liquid, warm, musical, sweet, enveloping", the more I think that the cans are not the issue. What you're writing describes the sound of a tube amp over a solid state one. The only tube headphone amp I've heard is the X-Can, which probably won't fit the bill, particularly with stock tubes. However, several highly regarded amps that might achieve what you're looking for include the Wheatfields, Earmax Pro, Berning MicroZOTL, and RKV, and perhaps the MG Head OTL. Caveat: I haven't heard any of these; my tube/solid state comparative listening has been restricted to the loudspeaker realm.