Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that Miss Kitty is drooling and that you are worried because her Rabies vaccine isn't current.
While Rabies can cause drooling it is much less likely to be the problem in an indoor cat with at least some history of vaccination. And Rabies tends to cause many other symptoms besides drooling in cats, including incoordination, and often aggressive, unpredictable behavior and possible seizures.
There are many other more likely causes of drooling in an older, indoor kitty.
Drooling can be related to eating something bitter (a bug or plant) or bad tasting, having something caught between her teeth, nausea, especially due to organ failure, or mouth pain from gum disease or bad teeth.
Because she is an older cat tooth and gum disease, a mass in her mouth, or organ failure with secondary nausea are other possibilities.
I would look closely into her mouth if you can to see if you can see evidence of redness, ulcers, broken or missing teeth, or anything caught.
If she has been sneezing and you see any redness or ulcers some upper respiratory infections, like Calici Virus, can also cause ulcers and these are painful, so they do cause drooling, and that is definitely a possibility in a cat that is excessively drooling and lethargic. Look for ulcers inside of her mouth, such as on her tongue and hard palate (roof of her mouth). If she has them the ulcers could be caused by Calici virus, which is an upper respiratory virus that can cause ulcers and joint pain as well.
Some cats get sicker than others and young, healthy adult cats seem to tolerate a respiratory infection and fight them off better than very young kittens or older cats. An upper respiratory infection in cats is just like a cold in you and I, and these are commonly caused by a virus. These are easily caught by breathing in virus particles in the air from a sneeze or nasal or eye discharge.
Because she is an older girl ideally she should see her veterinarian for an examination and some blood tests.
In the meantime you can help her feel better by adding warm water to her food to make it smell more (they don't eat if they cannot smell) as well as making it easier to chew and swallow.
Also the more fluids she gets the better. Offer tuna juice, low salt chicken broth, run the tap if she likes to drink out of the sink. With the amount of drooling she is doing dehydration is a real concern.
If she has any nasal congestion take her into the bathroom with you if you run a hot bath or shower as the steam will soothe her sore throat and airways. If her nose become very congested you can use sterile saline to loosen the thick mucous and remove it. She won't like it but it will help her breathe and be able to smell her food better. You can also use sterile saline to remove eye mucous if it accumulates.
You can use Chlorhexidene oral rinses (like CET rinse) on any ulcers or gum irritation to keep that area from getting infected by bacteria secondarily and keep them from becoming crusty and more uncomfortable.
If she is refusing to eat and seems nauseous to try and settle her stomach you can give either:
1)Pepcid ac (famotidine) at a dose of 1/4 of a 10mg tablet per 5 to 10 pounds of body weight every 12 hours
2)Prilosec (omeprazole) at dose of 1/4 of a 10mg tablet per 5 to 10 pounds of body weight every 24 hours
These are acid reducers and should help her feel less nauseous, and thus less likely to drool. These medications are quite safe and can be used long term if they help her feel better.
Then offer a bland diet of 2/3 boiled, minced, white skinless chicken and 1/3 boiled, white rice mixed with some low salt chicken broth to make it easy to lap up and swallow and get additional fluids into her.
If she runs a high fever (more than 104F), has a green or yellow nasal or eye discharge, stops eating even with coaxing and clearing her nose and eyes, or starts coughing or having difficulty breathing then she needs a veterinary exam as soon as possible.
If her extreme drooling continues then she probably needs fluid therapy from her veterinarian.
If this continues I highly recommend testing her for immunosuppressive viruses (feline leukemia and feline immunodeficiency viruses) as well as a complete blood count, biochemistry profile, and thyroid level. Immunosuppressive viruses will make a simple infection much worse as they stop the immune system from fighting infections the way it was designed to do.
Let me know if you have any further questions.