What is causing my car to run rough/die after fueling?

I own a manual transmission 1998 Toyota Corolla with ~202,000 miles on it. Whenever I put gas in the tank (and it does not matter whether or not I completely fill it), the car idles really rough and sometimes dies. If I sit at the gas pump and let it idle, then it will eventually even out and run like a champ (until the next time I stop for gas, of course). The "Check Engine" light is on all of the time. A mechanic told me that the gas cap might be faulty so, I replaced it. Unfortunately, it did not solve the problem. I would greatly appreciate your kind assistance. In the meantime, all the best!