That sounds unpleasant - I'm sorry you're so uncomfortable! It sounds like allergies to me too.
I'm allergic to lots of things, including hamsters
I've found that the only OTC medicine that works for me is Zyrtec. I don't think I'd be able to have a hamster if it didn't exist. It works for a full 24 hours (if I forget to take it at my usual time I'm reminded about 20 minutes later when I start sneezing.) Claritin doesn't work well for a lot of people - it probably wouldn't be a bad idea to try another OTC medication. Also, antihistamine eye drops can be helpful.
Allergy testing is wonderful (not the actual test itself - that's kind of awful.) I was tested a few years ago and learned that along with being allergic to pretty much every animal I'm also allergic to maple pollen...which explained why I'd wake up in the middle of the night with symptoms similar to yours whenever I left the window between my bed and the maple tree outside open. You could have a similar situation going on that testing will help you to identify.
As far as holding Aria goes, it's probably better to wait until the symptoms go away. You're most likely not sick, but it's not worth risking her health. Also, if it is allergies, holding her could exacerbate your symptoms even if you're not allergic to her (because your nose/eyes are already inflamed and possibly more sensitive to potential irritants.)
Feel better, and good luck figuring out what's wrong!