Dedicated circuit picking up noise

This may sound like a stupid question, but did you install
an additional breaker for the dedicated line in your breaker box?
Also, the ground loop you have may be an equipment issue and not a power issue.
If you have a ground loop in your system a dedicated line will not fix it.
As far as the ceiling fan, that's what prompted my first question. I don't have an answer for that.

Remember a dedicated line just guarantees no other loads on that circuit. But all circuits go back to the main panel and are ultimately tied together as are the grounds

You have two phase (hot) lines coming into your home that should be split roughly 50/50 to all your outlets. So the new new line could be tied to the same hot side of the two hot lines in your box. There is still the other line to try, that will still have your other half of the outlets tied to it also. And like mentioned above they all share the same neutral/ground at the breaker/service panel. So it's not uncommon for some of that noise to get in your dedicated line.

The ground loop must be something in your system causing it, assuming the new line was done right. Maybe different IC's might lesson the hum. If it is a tube system, a tube can be going bad. Cable TV is also a known problem if your system is tied to it for an audio/video system.

Try moving to the opposite phase at the breaker box and see if that solves the problem. If your ceiling fan has a re-stat speed control, it may be the source of the hum.