Like when you smell a dead rat your mouth produces saliva and makes you spit a lot. Or when you see something gross, doesn't make you vomit, but your mouth waters you spit a lot.
The condition you've described might best be called
si·al·or·rhe·a (sī-ăl′ə-rē′ə) n.
An excessive secretion of saliva.
(American Heritage® Dictionary of the English Language, Fifth Edition. S.v. "sialorrhea." Retrieved April 1 2016 from http://www.thefreedictionary.com/sialorrhea .)
According to the Farlex Partner Medical Dictionary, synonyms of 'sialorrhea' include "hygrostomia, ptyalism, sialism, sialismus, sialosis".
As far as I could determine, no verb form with 'sial-' exists.
An alternative is
pty·a·lism (tī′ə-lĭz′əm) n.
Excessive flow of saliva.
(American Heritage® Dictionary of the English Language, Fifth Edition. S.v. "ptyalism." Retrieved April 1 2016 from http://www.thefreedictionary.com/ptyalism .)
The verb, 'ptyalize', is an obsolete medical term.
If you want to avoid words that sound clinical or technical, one term used in the past was
sweet-spittle n. Pathol. an increased secretion of saliva having a sweetish taste.
["sweet, adj. and adv.". OED Online. March 2016. Oxford University Press. http://www.oed.com/view/Entry/195665 (accessed April 01, 2016).]
RETCH, verb intransitive [See Reach.]
To make an effort to vomit; to heave; as the stomach; to strain, as in vomiting; properly to reach.
"Heave" and "gag" fit fairly well, too.
"Drool" refers exclusively to the saliva production and dribbling, so also fits this use.
Now, as to the opposite of salivating would be dry/cotton mouth or even desalivating. Doesn't seem to fit your particulars, though.