Timothy hay is well known to cause allegies in people. I’m allergic to timothy myself, so I give my bun orchard and oat hay in the bedroom. He gets a little bit of timothy in the living room, because I’m mostly bothered when I sleep (I cough and cough endlessly). Hay is very variable, so some batches can be dustier/have more pollen than others. So it makes sense you react more to some hay boxes than to others.
I would advise you to try and steer clear of timothy hay. You seem to have a pretty bad allergy to timothy, so for the sake of your health (allergic reactions take a toll on the immune system), you should try and switch hay. Try to clean out every trace of the old hay as best you can from the bunny room. Wear a dust mask if you have to. Rabbits don’t absolutely have to have timothy, any grass hay that is good for horses is good for rabbits. Australian rabbits never get timothy hay because it doesn’t grow there and it’s very, very expensive to import. The only hay that should be avoided is alfalfa and clover (legume hays), it’s too rich for adult buns. It’s good for young, growing buns though.
i know we have one person here that’s doing monthly allergy shots, so far they’ve not seen improvement.