Can't use trackpad to look around while moving in my first person game

I'm creating a game that utilizes the First Person camera angle and just finished the fist game shell for it. When using a connected mouse, I can move my character and have it look around at the same time (no problem), but then when I try to use the trackpad on my laptop, I can no longer walk and look around at the same time. If I'm stopped, the Track Pad will allow me to look around, but as soon as my character begins moving, the trackpad no longer allows my character to gaze the scenery. I believe that Unity only exposes Input for known devices (keyboard, mouse, joysticks/gamepads) and maybe it doesn’t see the laptop trackpad as one of the Input devices for the “mouse” camera movement for looking around (though I'm not sure that would explain why the trackpad works while my character is at a standstill). Any ideas on what's going on? Thank you.