I saw enough people saying that Bluetooth helped that I just bought a set of bluetooth headphones, and yep, now works fine.
Ironically I bought the headphones from Amazon, which makes me feel like a bit of a chump considering the circumstances.
Audible are lucky they're pretty much the only player in the audiobook world. The way they're responding to customers is criminal. I asked them (several times) the direct question "Are you going to fix this? Because if you're not, I'm not going to continue to pay for a subscription that I can't use", and they act (every time) with complete surprise, like nobody has ever reported this problem before, and it will surely be fixed very quickly.