The main use is dégouliner de + noun but noun + dégoulinant(e) works too.
To me it means it's spouting profusely (but slowly), like in every direction and non-stopping, so much that it's nauseating. There is so much of it that it overflows then it trickles: ça déborde puis ça dégouline. You can use sentimentalisme débordant too, or déborder de sentimentalisme, and even mix them:
Ça déborde et dégouline de sentimentalisme.
I wouldn't say from one's mouth as a general rule, since you can say it about a book or a movie too:
Ce film dégouline de sentimentalisme, c'en est écœurant.
Note that like Laure noticed, "déborder/dégouliner de bons sentiments" is often used instead of "déborder/dégouliner de sentimentalisme", it's more idiomatic.
You could say it with any noun standing for a feeling, an atmosphere, even if I think it's mainly used to point out positive feelings and express there is so much of it it's becoming nauseating (so it's a bit cynic):
Dégouliner de gentillesse, d'amour, d'optimisme, de bonté...
I wouldn't say it's a mistake to use it with negative feelings (dégouliner de haine, d'horreur, de tristesse...) but you obviously lose the cynic touch since you are already talking about negative feelings and not turning good feelings into bad ones.