Yes it is possible if their saliva if there saliva is extremely thick.
A person can develop a mucus plug which cuts off their airway.
This is usually in a person that has a compramised airway already such as on a ventilator.
If you have a specific example let me know.
In a normal healthy person with thin saliva you don't have to worry about choking on your saliva.
The reason you are feeling way is that anxiety can make your throat feel tighter.
Are you taking any anxiety medications or seeing a therepist?
I can understand your worry about taking Depakote.
Was it prescribed for the anxiety or something else?