Your pet is a family member, not disposable. You love your pet and have made a commitment to care for this pet for a lifetime, so re-homing is not a good option. Of course you should speak with your conventional and holistic human health care professionals for diagnostics and treatment, but I do have a few suggestions.
You may be allergic to your pet’s fur, dander or saliva. Many believe cats are more allergenic because they groom themselves more, leaving saliva on their fur. Thus bathing them frequently can remove the saliva allergens. Yes, it is possible to bathe a cat. If you use Nature Rich soap, you can soak a pet daily. The coconut oil in the product is good for the skin. You don’t have to worry about drying out the skin. It is smooth and very soothing. Swaddle your cat and use a small water basin to soak your cat without exposing him to scary running water.
There is a lotion product called Allergen –C which you can use to wipe the dander off of your cat.
Feeding your dog or cat a quality fish oil on a regular basis can decrease shedding. Some guardians vacuum their pets to decrease fur exposure. Vacuum your home frequently and it may be helpful to not allow your pet to sleep in your bed. Your pet should be groomed frequently, but it may be best if a non-allergic family member does the brushing.
Diffuse essential oils in your home to decrease allergens and to support your immune system. One of my favorite blends is YL Purification.
There are many chiropractors in the area who utilize the Standard Process line of whole food supplements. Several of these products can be taken orally to support your immune system or to be used as sources of natural antihistamine to counter the congestion, redness or itchiness that you may experience when you snuggle with your pet!