This is not likely to be related to the bite and is more likely a couple of different problems that are commonly seen after adoption.
Cats often get upper respiratory infections in the shelter and when they get adopted. In most cases this is similar to a cold in people, but they have potential to get more involved.
Diarrhea is also quite common when cats shift to another environment and/or food.
In both cases, some sort of infection is present, but stress is often the initial cause. Most reputable shelters offer a time period (two weeks or so) where the adopted animal can be brought back for vet care. I recommend calling them at your first opportunity to ask about this and hopefully schedule an appointment with a vet. We can usually get these infections cleared-up with treatment. For more information about these conditions, I recommend you read the articles at the links below.
https://www.petcoach.co/article/most-common-causes-of-diarrhea-in-cats/https://www.petcoach.co/article/5-things-you-should-know-about-feline-respiratory-infections/I hope that helps and Est gets better soon.