My tube amp causes isolation transformer to buzz

There is a gage on the front that as you increase the voltage goes up. It is labeled in “volts” and has a green “safe” zone (approx 105-120v).
Is this what you mean?

Yes.

I don’t know what the accuracy of that particular gauge/meter is, of course, and personally I would want to verify the voltage with a good digital multimeter. But I’d feel confident that it is accurate enough for safe operation of the amp, although perhaps not accurate enough to be fully optimal in terms of sonics.

Good luck. Regards,
-- Al

The full specs on the power consumption of the Classic 60 are:

105-125VAC 60Hz (210-250VAC 50Hz) 500W at rated output, 700W maximum, 370W at idle.

I suspect that the 700 watt number doesn't reflect normal usage, but rather is approximately based on the current rating of the 6.25 amp mains fuse that is used in the design. (6.25 amps x 120 volts = 750 watts). And you are most likely using the amp's rated output power capability only rarely, meaning that the amp would be drawing significantly less than 500 watts most and perhaps all of the time.  So I suspect that you are not exceeding the 50% guideline Ralph cited above (corresponding to 400 watts for the particular Variac) by much if at all.

A basic question, though: Does this Variac have a built-in voltmeter? Or if not, how have you adjusted its output voltage?

Regards,
-- Al