My suspicion about the auxiliary adapter turned out to be the problem. Somehow the adapter was drawing power to the radio. I could unplug the adapter and the radio would turn off (with key removed), and plug it back in again and the radio would power back on.
I could not find any visible issues with the adapter, but I still removed it completely. I'll have to do some research on other brands available. I bought this adapter off Amazon for $25 last year - used it all the time.