british english - How to answer "Do you need a 5p bag"

I have troubles with this simple question: Cashiers do not understand my accent, so I want to find some answer that is easier to articulate.

When I answer " I already have one " - they never understand.

Is there any other option to say (apart of "No, thanks) that I have a bag (I brought it with me) that utilizes "better" vowels for non-native speaker?

British English UPD To those answering later: I want to improve my English pronunciation. The example in question is just one of the examples, not the whole issue, thus pantomimic clowneries could not be taken into account: NO, THANKS!

Answer

Cashiers want customers to get through the line quickly. They don't need justification. It is a simple YES/NO question. So "No" or "No thank you" to be more polite.

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David Richerby made an excellent point in his answer below that you need to consider head nods too. In English speaking countries a head nod up and down means yes, while shaking side to side means no.

in a comment on David's answer nigel222 made another great point. It is "No, thanks" but "Yes please". That gives the person you're speaking to two chances to hear you: "something please" implies Yes, "Something thanks" implies No. Useful in a busy / noisy environment, as well as polite.

I guess the other point here is that you need to be sure that you're speaking loudly enough. Checkout lines are noisy.

So saying "No thanks" with a couple of side to side head shakes should get you through the line without a "Huh??" from the cashier.