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Good morning, Sarah.
You're never wrong to take a cat to the vet so that you can get an unbiased opinion on what is going on. There are plenty of things that can cause open-mouthed breathing, but we certainly want to rule out allergies and an upper respiratory tract infection. If this was a recent adoption, the risk of a URI is increased as many cats are exposed to URI concerns in both shelters and rescues.
If you wish to try nasal drops, saline are a good choice and should not cause any issues. If you see that symptoms persist for more than 24 hours after starting the drops without improvement, your companion stops eating or drinking, etc. then seeing a vet will become imperative.
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