Chromatic embellishing chords are by definition never essential harmonies. As described by Gauldin in Ch. 32, they usually function as passing or neighbor embellishments to other harmonies.
Augmented Triads. Augmented triads usually supply a dominant function (primary or secondary). Gauldin says that you usually find augmented triads on V or I, but the I chord will almost always go to IV, making the I chord more accurately V/IV. Notate augmented triads with a superscript + after the V.
Augmented triads will have both the chordal third and chordal fifth resolve up by 1/2-step, to the root and third respectively of the resolution harmony. The augmented triad will often appear in first inversion, which heightens its use as a passing harmony.
Dominant Seventh Chords with altered fifths. These are standard dominant seventh chords, but the chordal fifth has been altered to create either an augmented triad with a minor seventh, or a diminished triad with a minor seventh. See Gauldin for his preferred notation (+ and ° are NOT superscript).
Common Tone Diminished Seventh Chords. Common tone diminished seventh chords do not function or resolve in the same way as Leading Tone diminished seventh chords. The root of the resolution chord becomes the seventh of the diminished seventh chord. Other voices resolve stepwise to chord tones. The common tone diminished seventh can function either as a passing or neighbor harmony. Gauldin denotes them as P°7 and N°7.
The N°7 appears between two instances of I or two instance of V. Double the fifth of the triad (before and after) to allow for stepwise motion in moving voices.
The P°7 creates chromatic motion in the bass. For example, from a I to a V43, or a I6 to a ii7. If part of a bass ascent, the common tone of the P°7 (the chordal seventh) is the root of the following chord. If the bass descends, the common tone of the P°7 is the root of the previous chord.
In actual music one often finds common tone diminished seventh chords spelled enharmonically to make individual lines easier to read. This does not change the function of the chord.
Common Tone Augmented Sixth Chords. These chord are closely related to the common tone diminished seventh chords. If you substitute b^6 for natural^6 of a common tone diminished seventh (between instances of a tonic harmony), then you end up with an embellishing German augmented sixth chord.
The NGer6th (Gauldin) occurs exactly like a N°7, between two instances of tonic. The PGer6th often prolongs a final tonic harmony in a quasi-plagal way, with the bass holding throughout, and chromatic motion in the three upper voices (see Gauldin p. 591).