Have you checked the f8lter for the air/heat? It is possible the filter contwins allergens triggering your symptoms. Just a quick thought.
Other things that come to mind, mold, mildew somewhere in the house, weather in he bathroom walls, foundation or directly outside somewhere . Dust, pollen up in the attic or in the air vents that can cause a reaction, the air and heat usually share some commen duct work.
As far as the cat, if you had a dsh before and replaced it with another dsh then chances are it’s environmental rather than the cat, if you want from dsh (domestic short hair) to dlh (domestic long hair), you could be allergic to the cat itself. Sounds funny but the coat itself you can be allergic to a short hair but not long, or you could be allergic to long hair not short, everyone’s different.
I found out that the short hair cats 8m allergic to but not curly, medium or long hair. With goats and rabbits the angora coats seem to suck all the oxygen out of the area, I can barley take a breath in after I’ve been with one for 5 minutes.
You can see your doctor and get allergy tested to see if your allergic to the cat but I suspect it is the house having some agent your allergic to weather it’s in the walls, carpeting, basement or bathroom is a trigger. Do you have these symptoms around other cats or just yours? Do the symptoms disappear as soon as you leave the house or do they stay with you once you’ve left?
Any bedding, pillow etc new or different? If you rule everything else out or the allergies only happen in one area etc you can pretty much rule it’s environmental vs the cat. You can try allergy tablets or medicines to be symptom free or you could get allergy tested by your doctor. Getting rid of the cat should be a last thing if you get no allergy relief but most people are willing to be a little stuffy over their pet or take allergy meds to relieve the symptoms.