Okay, this might be a bit long.

I adopted my cat from a foster family (where he was also born and lived for a year) about two months ago. When I got him everything seemed fine except that he seemed to have mild allergies (sneezing, watery eyes). 

The sneezing seemed to get worse and he was expelling colored snot so I brought him to the vet, who put him on antibiotics and said he probably just had a head cold. 

Shortly thereafter (then being about a month ago) he started having diarrhea. All the time. He had solid stool when I got him, I didn't change his food from what he had in the foster home, and he hasn't had a solid stool in over a month. So I brought him to the vet again. She diagnosed him with tapeworms and gave me some more meds for him. He still has diarrhea and now his cold symptoms seem to be coming back as well, he sounds like he can't breathe through his nose very well and constantly sneezes. 

Should I be alarmed about this? I plan to at least call the vet and get recommendations, but has anyone had this experience? Can cats just have horrible allergies? And should I try switching up his foods? He already eats food without grain in it. I'm not sure what I'd switch to, since all the advice I have found online assumes the cat is eating bad food to start with.
user
My cat has tapeworms, is diarrhea-prone,
assistant
Okay, this might be a bit long.

I adopted my cat from a foster family (where he was also born and lived for a year) about two months ago. When I got him everything seemed fine except that he seemed to have mild allergies (sneezing, watery eyes). 

The sneezing seemed to get worse and he was expelling colored snot so I brought him to the vet, who put him on antibiotics and said he probably just had a head cold. 

Shortly thereafter (then being about a month ago) he started having diarrhea. All the time. He had solid stool when I got him, I didn't change his food from what he had in the foster home, and he hasn't had a solid stool in over a month. So I brought him to the vet again. She diagnosed him with tapeworms and gave me some more meds for him. He still has diarrhea and now his cold symptoms seem to be coming back as well, he sounds like he can't breathe through his nose very well and constantly sneezes. 

Should I be alarmed about this? I plan to at least call the vet and get recommendations, but has anyone had this experience? Can cats just have horrible allergies? And should I try switching up his foods? He already eats food without grain in it. I'm not sure what I'd switch to, since all the advice I have found online assumes the cat is eating bad food to start with.
user
My cat has tapeworms and has
assistant