The baby is watering.

[mampwamp]

The baby is watering.
The baby is drooling.
The baby's saliva dropping on my hand.

were those sentences above correct?

 

[SoothingDave]

The second is. Babies drool.

 

[Matthew Wai]

The example below may be helpful.
"Watch out, the baby is dribbling on your shirt!"── quoted from http://www.ldoceonline.com/dictionary/dribble_1

Not a teacher

 

[emsr2d2]

In BrE, "dribbling" is more common. "Drooling" is used though.

 

[Roman55]

I am not a teacher.

Animals and dirty old men drool.

Babies and footballers dribble. ;-)

 

[Barb_D]

In the US, babies drool.

 

[Tdol]

The baby is watering.

This one doesn't work. We can say that someone's eyes are watering, but not that the baby's watering.

 

[MikeNewYork]

We also say "the baby is spitting up".

 

[emsr2d2]

For me, that means that the baby had produced a little bit of vomit (usually some undigested milk). Dribbling is just saliva.

 

[MikeNewYork]

Yes, but if it is juice, it can be difficult to differentiate. That said, I am not a baby expert. ;-)

 

[Rover_KE]

There's also slavering (with a short 'a').

[I took your title to mean 'The baby is wetting its nappy (diaper)'.]

 

[Matthew Wai]

Is it correct to say "the baby is suckling and dribbling milk"?

 

[emsr2d2]

I connect "suckling" with animals. I would say that a baby is "breastfeeding" or just "feeding".

 

[Tdol]

Suckle for babies sounds old-fashioned to me. I would use it for animals too- suckling pig, for instance.