I think BILD didn't choose the correct word. (Oops, incredible: BILD wrong?)
BILD writes about the Frankfurt city parliament. Therefore the correct word should be Babbelwasser. The phrase "Hast Du Babbelwasser gesoffe?" is typical Hessian (and especially "frankfurterisch"). A person who has "Babbelwasser gesoffe (= getrunken)" does not stop talking.
I guess the author of the article does not come from Frankfurt (or Hesse), otherwise he would not have used "Sabbelwasser" which is a foreign word for local readers.
An objection raised in a comment is why BILD which has a nationwide audience should use local (Frankfurt) dialect. The reason is that BILD has 20 regional editions ("Regionalausgaben") presenting news about regional topics. To read such articles you must intentionally navigate on the homepage to the corresponding region. The link given in the question has the URL
which shows that the article is addressed specially to local readers. Its headline is "BILD sagt, wer im Römer richtig ranklotzt". Römer is the name of the Frankfurt city hall and I guess already that is not known supra-regional. So, why should people in Berlin or Munich be interested in local Frankfurt politics? And why should the typical local expression "Babbelwasser" be replaced by "Sabbelwasser" which probably nobody knows in Frankfurt?
Moreover, in my opinion the word "Sabbelwasser" may be a bit ambiguous. It is certainly derived from the verb sabbeln which can also be used as a synonym for sabbern. The verb babbeln does not have the ambiguity.
The word "Babbelwasser" even has an occurrence in serious literature. In the novel "Das siebte Kreuz" by Anna Seghers (a Mainz native) we can read on p. 59
»Wirklich, du bist so weich in der Ehe geworden, Auguste«, sagte Ernst, »du warst mir früher zu kratzbürstig.«
»Du hast schon in aller Früh Babbelwasser getrunken«, sagte Auguste.
Here it is used in the sense of talking nonsense.