sneezing & sicky cat

I have a birman monster, who since sat keeps sneezing. he's an 11yr old rescue cat. he always has a dodgy tummy and is frequently sick and has the runs. He is on dry food which controls things but since sat has refused anything but wet food, which luckily he's keeping down at present and no unexpected suprizes left on the floor. He's fully vaccinated, but is not quite himself and of course sneezing. My tabby was also sneezing last week but has now stopped and our other cat's fine.
Could he have a cold and if so what can i do for him?
Any advice would be most welcome-Thank you

Cats don't really get colds - they get upper respiratory infections or rhinotracheitis. Your cats need to be seen by a vet and get appropriate medications.

The fact that he is sick and frequently has the runs needs to be looked into as well. He could have worms, or other parasites.

Did you adopt this cat at 11 yrs old? Has he been checked out thoroughly by a vet?