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There's a USB micro-B socket under a small removable cover on the Axxess adapter, you can use an Android phone with USB-OTG and their app or a laptop to run a firmware update on the adapter, might be worth a try. Also lets you modify the mapping of the steering wheel controls if you've got them.
I'd pull the stereo out and hook a multimeter (or a 12V lightbulb, or something) up to the black+yellow and black+red pairs to check the output from the adapter is what you expect. I assume the standard behaviour is like most of the VWs where the power to the stereo stays on until you pull out the key, not just when you turn it off and that's what the Axxess adapter is trying to replicate.