I agree with Bam. Using a different type of hay, like orchard grass or oat hay, can be a great way to keep healthy hay in your rabbits diet without having to suffer as much. I couldn’t figure out why I was wheezing with a raspy voice for the first month of having my rabbit. I was worried I might be allergic to him! But it turned out I am allergic to timothy hay, too.
After switching to orchard grass I’ve had a much better experience. Yes, it’s true that people who are allergic to one type of hay are probably going to still have some kind of reaction to other types of hay, but orchard grass and oat hay tend to be less dusty and fragrant. It’s like choosing the best of the worst for hay allergy sufferers.
Beyond buying different hay, it will also help if you store the hay away from where you sleep or spend most of your time, if possible. I keep my extra hay in it’s original ventilated bag in the closet. If I keep it out open in the living room I can feel it tickling my nose and throat sometimes. I do think moving her to another room would be the most ideal for your health, but if you can’t do that, I would suggest getting a hay bag with a zipper top closure (instead of an open hay rack or bin). The zippered-top bags keep hay inside, so less cleanup, and reduce the amount of hay dust and allergens exposed to the air. I bought one on Etsy. Also, try to wear gloves when handling the hay and always wash up afterwards to keep from touching your eyes and face later.
Hay is the trade-off we make for having our rabbits but it is possible to manage. Good luck!