Before trying anything else, supposing you use Windows 10 (applies to 8/8.1 too), press Shift while clicking Shutdown your computer. Start as normally and check if the issue has gone.
Since it's affecting AH only, logically it should be a Settings issue.
To me it sounds like the stick is acting more like a button instead of a potentiometre. It reminds me of the issue I used to have with the view button: Just a tiny nudge instead of the usual movement. The culprit was that I had mapped an unorthodox mixture of buttons and pots or something like that. It might have been the rocker pot on the throttle being used for looking down left-right or something similar. Although renaming the Settings folder should take care of that, double check the mappings both for the stick and other controllers.
There's also a Stickdef folder within AH3, containing similar .jsm files as the Settings folder. Try renaming that too.
If none of the above helps, reinstall AH over the existing game.
