Max is going exactly where I am thinking as well. I am completely unfamiliar with the Bottlehead, so I'm at a disadvantage there. But I would suggest two things:
1. Make sure you are using quality interconnects. In a pinch, I recently went to Home Depot and purchased a 6 foot set of interconnects for use between my Fisher preamp and power amp -- only to have it hum notably. Swapped the interconnects out with a different set -- no hum. What I found was that cheap cables today do not even remotely wrap or shield the inner conductor but rather, simply run a bare ground lead next to the inner conductor, and count on the very low output impedance of today's modern SS devices to reduce any hum pickup to minimal levels. So, make sure you're using good, well shielded interconnects.
2. Taking Max's suggestion even further, using a known good set of interconnects, plug one end of the interconnects into the Dynaco, and at the other end, bring them together face to face, so that the outer ground shield of one plug is grounding out the top of the other plug, and visa versa. In this way, the top of both inputs will be grounded out by the ground lead of the other channel.
If the Dynaco is quiet in this test, then the problem must lie within the Bottlehead, and particularly with the grounding circuit between the output jacks of the two channels. On the other hand, if the buzz and hum are present, then the problem may be the inherent issue with the physical lay out of the Dynaco due to the power transformer being mounted in between the two amplifier input jacks. This allows the input and ground circuits for each channel to be on opposite sides of the magnetic field generated by the power transformer, which then generates a ground current through the interconnects when their ground leads are connected together at the source end. This is often a problematic way to do things, but normally, the ST-35 is not cited for this kind of problem -- but that is also based on it being used in conjunction with a Dynaco PAS preamp. Most particularly however, note that to minimize such concerns, Dynaco did not connect the ground terminals of the output jacks in the PAS directly to ground, but to ground through 10Ω .5 watt resistors. These resistors act to allow an appropriate ground reference to be made, but also minimize any ground current that might flow from the ground loop conditions described. You might look at using these resistors in your bottle head if the suggested test indicates that the noise is originating from the Dynaco.
As Max said, this problem is all but eliminated in the ST-70, since the input jacks are spaced right next to each other in the middle of the power transformer's center line. This kind of configuration allows for little if any ground currents to be developed between the channels, all but eliminating the problem. However, just to be sure, Dynaco included the 10Ω resistors inside the ST-70 as I recall as well.
I hope this helps -- let us know!
Dave