For eye gaze, are you using the web cam? I find it useful, but not that accurate. Eye gaze has "snap eye gaze" which locks the eyes to 9 positions (up down left right and combinations thereof). This stops eye jitter. If you turn this off, I recommend disabling camera input under Eye Gaze and use the mouse/touch input instead. You get much better control over the eyes that way. (This is personal opinion - but I have had times I am looking directly at the screen and the eyes bound around a bit - not a lot, but enough to feel uncomfortable watching it.)
Mouth positions (from the Face behavior) are controlled by sound, except for smile and surprise (they are from the web cam). So a well lit face for "lipsync" probably won't help. I would make sure the recording level is high enough etc. One way to test this is to do a recording and see how high the waveform is. I typically use Adobe Audition instead of recording inside Character Animator, because it has a "Normalize" effect to normalize the amplitude (makes it louder if needed). I then load that sound file into Character Animator and use "Compute Lipsync from Scene Audio" in the menus. The other thing you can do (for testing) is after recording a dummy lipsync, you can right click in the timeline and add your own visemes by hand. Its tedious and boring, but you can then make sure the mouth positions work correctly. If adding by hand looks good, then I would guess microphone volume problems as the most likely cause of sub-standard results.
But the tools are not perfect. Good, but not perfect. They do play up at times.