I've been tested but I knew before that because it triggers my asthma. In some houses, I can walk into and know they have a cat before I even see signs of it.
Based on your symptoms, it sounds like you are allergic.
My over the counter allergy medicine of choice is zyrtec. But my issues are so bad that I have to take extra medicines, have my inhaler ready, and be prepared to leave 10-15 minutes after arriving if the allergen level is high at the destination. For me, avoidance is really the best way to go.
I have other observations and suggestions that do not involve medicines, however, they only help the situation not fix it. I've found that places with hardwood floors are easier on my cat allergies than carpet. If you are visiting a person that has cats and they are willing to sweep or vacuum before you arrive, they should do it hours before you arrive so the dust settles. If the weather is nice, leave windows open and try to sit near one so you can get some fresh air and less of the room filled allergen air. Avoid sitting on cloth sofas or chairs where the cat dander may be hidden.
In my experience, the cats always know which people are allergic and seek them out. :-)
For a more permanent solution, you can try allergy shots. Check with an allergist about effectiveness. I think it may be different for adults than children. Also, allergy shots are a long term investment. You start going multiple times a week and work toward once a month, then less frequent from there. Some people may never be able to fully stop getting the allergy shots.