Our cat is a bit of a dribbler too. He is a rescue cat so we don't know his history of course he took immediatley to my hubby and whenever he is being stroked or cuddled or even just being talked to, he drools like mad. It did not happen at first but it seems that he adores hubby and just cannot contain his excitement. Now he drools on all our frinds too... Vet says No Probs with teeth and not to worry.
Hope your puss is OK sure she is pretty normal. All cats are slightly bonkers...