A few years ago I consulted a pharmacist about what I thought was a chlorine allergy. He told me that it was not an allergy but that chlorine was an irritant and advised me to wash out my sinuses with water after swimming. I tried this and also the Indian practice of neti dhauti (rinsing the sinuses with salt water) and these certainly helped, but I have found that the best solution is a nose clip. At first I disliked the nose clip intensely but gradually got used to it and sometimes I almost forget that I am wearing one.
I never wear a nose clip when swimming in a competition and because the races do not last very long, as a rule, and perhaps because the competition pools have a better quality of water than the pool I train in, I do not suffer severe after effects.
Much of the problem with chlorinated pools is the build up of chloramines, or so I read on the Internet, formed from the interaction of chlorine and organic substances in the water.
Have you tried a nose clip, and/or nasal irrigation?