A-ha, now I might have just the ticket. I've become more involved than I ever wanted to in esoteric electronic things like this...
Check muting transistors before the output jacks. In my experience, although this is not much reported online, these can fail often and for the slightest (unknown) reasons. This can then cause distortion, bleed between adjacent channels (on the same muting control line), and possibly other problems. I've fixed 3 synths with distortion (one twice!) by replacing the transistors, and 2 also exhibited some bleed-through before that. None were the AN1x, but I suspect it uses a similar system to mute unwanted audio during power on/off.
The transistors I've seen/replaced for this have been the now-discontinued 2SC2878, specialised for audio muting. They can still be found on eBay - get 10 or so while you can, but please don't hoard them away from everyone else ;-)
The other option I believe is simply to remove the transistors (desolder). However then make sure you don't turn on your speakers/whatever the AN1x is input to until it has been on for a few seconds, otherwise I'm not responsible for damage to other equipment due to possible DC or other noises during power on/off.
Bring back relays, that's what I say........
Let me know how it goes as I'm very interested to hear whether anyone else has the same experience.