I can only move diagonally up-right and down-left while firing. While not firing, or firing with the mouse, I can move in all directions. Is there a way to fix this (besides just using the mouse to fire, which I'd rather not do but will if nobody has a solution)?

It's a keyboard issue.

Workarounds, which may or may not work, are:
* Remap the controls. You might find some control mapping that avoids this issue for your particular keyboard.
* Try a different keyboard. Different keyboards behave differently. (For example, a keyboard might only give you the up-left and down-right movement options while firing.)
* Use a gamepad. Gamepads do not have the same issue with multiple simultaneous keypresses that keyboards tend to have.
* (As you've observed, using a different input device, like the mouse, to fire can also work.)