Hey there, this is Dr Linda, a small animal vet with 10 years of clinical experience.
It sounds like your cat is suffering with chronic gingivostomatitis, a condition that causes severe gum inflammation and oral pain.
The cat's immune system over reacts to the plaque that will naturally be on the teeth. This causes inflammation, pain and trouble eating.
Common signs will include drooling (as in the case of your cat) and possible also face rubbing, reduced appetite, bad breath and weight loss.
This condition can be brought on by a viral infection (such as FIV or FeLV) or can occur for no obvious reason. It is sometimes genetic.
For some cats, we can manage them with frequent dental cleanings under anaesthetic, daily tooth brushing, oral enzymatic cleaners and antibiotics/steroids as neeed. However, this treatment method is not always successful, especially in those who are severely affected.
Dental extraction of all of the teeth (or all of the teeth except the incisors) should be considered when medical management does not work. When performed correctly, cats are almost instantly pain free and their quality of life is much improved. They can eat much easier than before, as there is no associated pain. They can even eat hard kibble!