Have you tried seeing if there's a difference when running off just the battery (without shore power), rather than when hooked up to power?
If my suspicion is correct, and it's due to noise from the 12v power supply, then you wouldn't get the same symptoms when running only off battery (since you wouldn't be using the power supply then)
Try that and let us know what you find.
If that does make a difference then it's possible that there's a fault with the power supply (likely a blown electrolytic capacitor on the output stage), that you'd need to get fixed
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