This is happening only on Dolphin? This happen to me when the Steam app is in background and the control options is marked on "Xbox Controllers" on Steam options. When this option is marked, the steam simulates the mouse+keyboard on the controller, so the right stick is the mouse, ZR is the left click, ZL is right click. Maybe it's your problem, see if the option is marked. Go to Steam menu > Settings > Controller > General Controller Settings. A big picture menu will open, uncheck all the "Configuration Support" options ("PS4, "Xbox One", "Switch Pro" and "Generic" options), close this window and click on Ok. If it's already unchecked, or the Steam is not running, maybe there's another app that simulates the mouse+keyboard on gamepad. Hope it can help.