The vet examined the one cat, found nothing during the exam. He prescribed antibiotics and non-steroidal anti-inflammatory (Metacam). Said if not better in 2 weeks, to bring them back for overnite stay to be scoped.
Might this condition be related to hairballs as both occasionally have hairballs; however, this hard swallowing has been constant for several months now.
Cats certainly can develop sore throats. Sore throats, voice changes, hard swallowing, drooling, and reluctance to eat hard foods can occur for a number of reasons. Infections, immune system issues, foreign objects in the throat, polyps or masses in the throat, and certain dental problems are common causes.
Since both of your cats developed symptoms at the same time, I am most suspicious of an infectious cause. The infection could be caused by a virus or a bacteria that has infected both. If your cats are related to each other, then simultaneous development of polyps or immune system issues is possible. An allergy to a new product or substance in the house also might be involved.
A course of antibiotics is not completely unreasonable in this situation (although some academic types frown on using antibiotics unless a known bacterial infection exists). However, the manufacturer of Metacam recently issued a warning against using the product more than once in cats. I recommend that you stop the Metacam immediately and talk to your vet about this.
Performing laryngoscopy (visual evaluation of the back of the throat) is a good idea. If no obvious polyps or lesions are present, then I’d recommend swabbing the back of each cat’s throat. The swabs can be submitted for bacterial culture and for viral (and bacterial) DNA testing through PCR (polymerase chain reaction).
Meanwhile, feed soft food and stop the Metacam!