TLDR; Hold down key opposing motion camera is move towards to stop the camera, then while keeping that key pressed Alt+Tab out of X-Plane. Release all keys, and Alt+Tab back into X-Plane. From this point, use arrow keys to move around.
For those who find themselves with this bug, I discovered a solution that will at least you finish your flight if you can't restart your sim. In my case, my camera kept moving to the right (D key), so I held down the opposite direction (A key) to stop the camera movement. While holding down that key, you alt-tab out of x-plane. Release the key, and then alt-tab back into X-Plane and the camera should stop moving. However, there is a caveat: you can't press any key in the direction that was affected by the bug (A and D or W and S). If you do, the bug triggers again. I found it best to stop using WASD at this point and just use the arrow keys for the rest of the flight.
The best method to solve this is to save the flight in the airframe manager and rload X-Plane, but the above method helped me when I was on approach on VATSIM.
