When you say "swing", are you talking about the "pendulum effect"?
BTW- my AXe CP drifts left as well.....but if you just gun the throttle just a bit, you kind of just bypass that and can get to a more stable zone(so to speak).
My big problem is that annoying "throttle cut" at about 60%(i estimate)......heli just slams into the ground......very hard, then bounce back into a hover.This,in my opinion.....SUCKS.
Why does it do this? Does on sim as well!!!!!!