Generally the converter humming is a sign that the converter is energized and working (and old), though it's possible that there's some discontinuity between the converter and the batteries. The converter shouldn't be a loud hum, though.
When the converter is plugged in do the 12v
lights and devices work in the trailer?
If memory serves, some Excellas had a switch to turn off the converter, but I don't think Airstream had started putting in the battery-disconnect switch by then (I may be wrong about the disconnect switch.) Newer trailers do have a store/use switch that disconnects the battery via a solenoid, and some previous owner may have installed a manual disconnect in yours.
You can have the batteries checked out by someone with a tester, or you can try charging them from a known-good source and see if they recover. If something is wrong with the converter (and they don't hum loudly or interfere with the radio when everything is working properly) it may have been over-charging the batteries, or it may be as simple as a broken connection or bad fuse between the converter and the batteries.