I wouldn't resolve an augmented triad by "lowering its root by a semitone" (*). Apart from anything else, totally symmetrical chords such as the augmented triad don't have a root. (All of the notes in the chord are a major 3rd apart, hence the symmetry).
Most commonly, the augmented note (G#) in your example, is treated as a leading note, so C-E-G# typically resolves onto A major (or A minor).
However, like its symmetrical friend, the diminished 7th chord, any of the notes of an augmented triad can be treated as a leading note. So, if you want to treat C as a leading note, your augmented triad could resolve onto a chord of Db major (or C# minor). If you want to treat E as a leading note, your augmented triad could resolve onto a chord of F major (or F minor).
This flexibility of treatment gives the augmented triad the ability (like the diminished 7th) to create a modulation to a variety of unrelated keys - although both chords are more commonly used for chromatic effect rather than modulation.
Does that help?
(*) Perhaps that "lowering the root by a semitone" refers to the bass in your example moving down from C to B while the upper parts continue with E and G#. This doesn't really resolve the chord as the B would simply be a passing note. It would, however, give you a second inversion of E major which would then resolve by moving to a chord of A (major or minor).
