Woodland Scenics Water Effects is a thick gel that dries transparent and can be roughly shapped. If it needs to be freestanding, you might need to use a core of fishing wire or cut clear plastic as thin as possible from a piece of blister. It would also work for something dripping down the chin that's supported. Just get a small glob on the end of a toothpick and 'draw' lines where you want them. Once it sets (very small streams like that will set quickly), you can go back in to add bumps and droplets and thicken it up in places. I've also used 5 min epoxy at around minute 4 to make icicles. Saliva might be possible, but I think the Water Effects would be a much less irritating way to do it.
I used Vallejo water effects and the clear thread used for making necklaces, cut a suitable length, and apply the effects till you are happy, when it's dry you can also color it if you wish
UHU is just a clear general purpose glue, but not a PVA, for bonding paper, card, rubber(I think) those kind of things
Do you have some thing like that there?