I am sorry to hear that Riley is drooling and has symptoms of an upper respiratory infection.
Drooling can be related to eating something bitter (a bug or plant) or bad tasting, having something caught between her teeth, nausea, or mouth pain from gum disease or bad teeth.
If she were 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 redness, ulcers 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.
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.
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 and thin mucous, making it easier for her to breathe.
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 the ulcers to keep them from getting infected by bacteria secondarily and keep them from becoming crusty and more uncomfortable.
Some lethargy is understandable, let her rest as she needs rest to get better. If your cats normally go outdoors keep her (them) inside until they are back to their normal playful selves.
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 an emergency veterinary exam.
If her extreme drooling continues then she probably needs fluid therapy and pain medication from her veterinarian.
Sometimes these upper airway infections turn into pneumonia so that's what we need to guard against. In most cases antibiotics aren't needed and can contribute to a decrease in appetite so I don't tend to prescribe them unless I feel there is evidence of a secondary bacterial component. These include a green or yellow eye or nasal discharge, evidence of pneumonia upon listening to their lungs or an infection that lingers beyond the normal 7 to 10 days.
I highly recommend testing her for feline leukemia and feline immunodeficiency viruses since she is a rescue. These 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.
Here is a link if you want to read more about Calici virus: http://www.pet360.com/cat/health/calicivirus-in-cats/42gJj403b0idm8v6BjfOiA
Let me know if you have any further questions.