Is it just a chlorine filter, or does it also filter out VOCs, heavy metals, chloramine (a different compound from chlorine) and things like iron rust and stuff?
Have you checked to see if your city fluoridates its water? Fortunately mine doesn't, but if they do this could also explain why you feel so bad afterwards.
I remember feeling that same way, along with a throbbing headache after every single visit to the dentist when I was a kid. Now I realize it was fluoride poisoning from that toxic garbage they put on your teeth.