LPC produces unexpected residual
It sounds like you're doing things correctly.

Think of LPC as a compression method. A lossy method at that, if you discarded the residual and replaced it with a synthetic pulse train (which might consider using for pitch shifting).

In a perfect world where the LPC coefficients were a perfect match to the formant filtering applied by the throat/mouth/nose, the residual would be exactly the glottal impulse.

However, since the LPC is just an approximation, the residual contains additional information.