Do you have a mental picture of how any transformer works?
Basically, one has two separate coils of wire that are next to each other, but also electrically insulated one from the other.
When AC is applied to one coil of wire, the magnetic properties of electricity induce a flow of electricity in the adjacent coil.
If the number of windings on each coil are the same, 100 volts into one coil gives you 100 volts out of the other. If the number of windings on the second coil is double the first, you'll get 200 volts (or, if you switch which coil has double windings, you'll get 50 volts out).
The latter example is what lets a high voltage, low current vacuum output tube power a low voltage, high current speaker.
Note that DC (direct current) will not pass through a transformer. This makes a transformer useful for "isolating" electrical devices from the main AC circuits. This can address ground hum problems. Transformers also have an upper frequency limit for the AC they pass, so depending on design, they can be useful for addressing RFI problems.
Transformers also have a maximum load they will carry. It sounds like you've discovered this already.
My only caution about combining devices is I'm of the school that one shouldn't spend time fixing problems one isn't having. You can end up creating new issues. If noise from your AC is an issue, address it, and then when you've got it fixed, stop working on the problem.
