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.

There is no link with the mouth.

In fact, it is related to a movie/story judgement. When everyone is good, when autor is only using scriptwriter trick to raise good feelings (be kind with old people / feed poor people / be honest / sacrifice …), basically when there is a tragic music when you have to feel sadness… it is not comparable to real life, to something “vif” (something trully alive) and to something fluid (something not simulated, natural beeing).

You find it, in French, fluid has the liquid meaning and this particular meaning of not forced (not faked). When there is too much feelings, it is not anymore appreciated and fall into the overflow.

It is why you can find lexical field of something “too much”. And there is so much than it brims over and flood of terrible taste.

So terrible taste than it causes us to “dégueuler“. (you can find the same root of dégout).

I might add it could also just be like something wet which is dripping and not necessarily gushing, and this can conjure up many images. You’ll find many examples such as the drop of water on a leave (la goutte d’eau dégouline sur une feuille), the candle (wax; la chandelle dégoulinait), rain (la pluie dégoulinait), sweat (je dégoulinais la sueur), etc. It can be messy, but there’s no foaming of the mouth here as this is not about anger nor drooling. The judgment call is about what is perceived as an excessive focus on emotion, it’s not about characterization of the emotional response the person has. I would think that this sentimentalisme dégoulinant can be described as a tearful, maudlin, sickly sort of cheap corny/slimy sentimentalism which leaves stains, and which is sort of overpowering (gravity). It is often marked as colloquial, by the way.

Concerning other nouns which would typically be associated with the adjective dégoulinant, the French Books corpus can help you with that (ngram); it yields: sang (blood), visage (face), sueur (sweat), eau (water), front (forehead), pluie (rain), corps (body), cheveux (hair). There is no surprise there. You did not ask what adjectives usually follow sentimentalisme, and this request yields: religieux, romantique, politique, larmoyant (tearful), humanitaire, exagéré, allemand, vague, mystique. Generally no surprise there either.

Le sentimentalisme excessif, est considéré comme quelque chose de tellement “sucré” qu’il en est écœurant.

On retrouve ce sens dans l’expression “mielleux”.
Ex:
Cette personne est tellement mielleuse, que je ne la crois pas sincère.
Toutes ses paroles mielleuses me débectent.

On parle aussi de “guimauve”.

On imagine très bien comment ça pourrait dégouliner de guimauve…
C’est l’image qu’on a dans la tête quand quelqu’un parle de “sentimentalisme dégoulinant”.
On pourrait dire: Tout ce sentimentalisme, c’est vraiment de la guimauve !

On retrouve l’idée de:

• Trop-plein qui déborde.
• Quelque chose d’écœurant tellement c’est sucré et collant comme dans l’expression “mielleux”.