Your headphones being balanced is immaterial here, so you can take that off the list of things too worry about.
The first thing to do is find out if it really is a ground loop. The most reliable quick way to do that is to grab a handful of ground-lift/cheater plugs (they’ll be about $1.50 at the hardware store), and add one to each piece of gear until the noise goes away. If that process fixes it, it’s a ground loop and then you can start looking at how best to address it for your particular setup without resorting to lifting one or more grounds permanently.
That it was present at a low level in one room, and is much more pronounced in another, makes me wonder if its not down to a cable TV box, and if that is the case (e.g. the above doesn’t fix it then it remains a possibility) you’ll need a cable TV isolation transformer.
Adding a balanced amp will not necessarily fix the issue.
Even in fully balanced systems you may need to add a DI box, or similar, (which is another type of isolating transformer, this time operating on the actual signal connections between gear), but those have their own issues since you’re adding transformers and additional cabling in the audio path.
Fancy power conditioners won’t always fix the issue either … nor will the “Hum X”.
So, first, do the ground-lift test and see if it really is down to a true ground loop and we can go from there.