It's maybe my gashness, but I'll be honest in that I have never responded to an "avoiding action" instruction by disconnecting the automatics as this would (I think) increase, what can become, a high workload situation. Likewise, I've never seen anyone else do it.
Wow , how many "avoiding action" instructions have you received? I've got around 12000 hours in busy airspace and I've never received one. If I ever do I hope to follow my company SOPs and disconnect and fully expect my FO do the same!
Hopefully you are confusing a routine turn from ATC to ensure separation with a full on "Avoiding action!..." call. Time to get your head into that manual and clear that one up before your next flight perhaps.