You can use a X-box One/xbox 360 controller in STO on a pc. Look at the xbox 360 tab at the top of the Feedback>Controls and User Interface page for information on how to setup the X-box controller.
I use an Xbox One controller to move my ship I have the same problem in combat my ship doesn’t move with the controller, or mouse, or keyboard. This happens on all of my characters ( lvl 60 to TOS). This started the day AOY was released @maxworf56 I use the following text file x360_spaceREV4 to use the controller.