First, an alliteration is not a repetition of consonants, as Jason Baker has written, nor is it the repetition of the same letter or sound at the beginning of consecutive words, as Mark Baker has written, but an alliteration is the repetition of the same sound (not letter) at the beginning of the stressed stem syllables of words (consecutive or not).
For example, "silken sad uncertain" (three /s/) in Edgar Allen Poe's The Raven is an alliteration, despite the "c" being a different letter and not at the beginning of the word, and "I have stood still and stopped the sound of feet" (three /st/) in Robert Frost's Acquainted with the Night is an alliteration as well, despite the non-alliterating "and".
The example by Robert Frost also shows that to say that an alliteration is the first sound in the stressed stem syllable is not quite correct, because it is the onset of that syllable. The onset is the first consonant or group of consonants (an onset cluster). (If the syllable starts with a vowel that is a "null onset".)
And that explains part of the dissonance in your example. "Sweetly slimily**, and softly" is not an alliteration, or not a pure alliteration.
The other part of the dissonance comes from "slimily" having three syllables and breaking the rhythm of the sequence. "Sweetly, softly, and slimily" flows a bit better, maybe, but the alliteration is still broken and incomplete.