Yes, it worsens some allergies but I do love a good rain to take summer pollen out of the air! I've had allergies -- including pollens and dust -- all my life and cleaning/dusting/laundering more after our cat's asthma diagnosis has reduced them more than just about any medications ever have.Humidity worsens some allergies like dust mite allergy.
In Adam's case, my first suspicion would definitely be the diet because of his diarrhea. But inflammation can come from lots of places so it's always worth reducing dust and pollen in the cat's (or human's! ) environment.