The mistake you've made (and even if you were a native speaker this is a minor mistake), is to answer a yes/no question with a sentence that the cashier needs to fully understand in order to work out whether to sell you a bag or not.
Simply change, "I already have one" to "No thank you, I already have one", make sure that you can enunciate "no" in a way that's clear to natives, and you should be good to go.
If the cashier hears "no something something something", and they ask you to repeat because they want to know what the "something"s are, then you might have to cut it down to just "no thank you". But if you start out by answering the question then the pressure is much reduced for both of you, because the urgent and important part of the communication is done.