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.

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.

A simple "No, thanks" should be understandable in almost any accent; "no" is less polite but isn't rude and should be even more understandable.

In the UK (and most but not all other countries), the standard gesture for "no" is to shake your head, i.e., to turn it slightly to each side a few times. If you shake your head while saying "no", things should be very clear. However, there are some cultures, mostly between the south-eastern Mediterranean and Iran, where a single nod of the head (moving it slightly up but not down) is used to mean "no". In the UK, and most other countries, nodding up and down means "yes". If you're from a culture that uses a small nod to mean "no", it's possible that you're saying "no" verbally while making a physical gesture that looks like yes. That would be very confusing to a British person: you'd be saying "yes" and "no" at the same time.

The mistake you've made (and even if you were a native speaker this is still 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.

"No, thanks". No other information needed, and in a busy shop it slows things down. "Yes please" if you do want one.

For readers not living in England it is perhaps worth pointing out that the 5p bag charge was recently introduced, and so there's no long-established protocol for this small negotiation. I recently passed my own previously used carrier bag, bearing a different store's logo, to the sales assistant in a clothing shop so she could fold my new clothes into it. She was briefly nonplussed (a "what's that" expression on her face) before she caught on.

In my experience the most easily understood response is the most predictable response.

In Singapore, you say "no need" or "no thanks".

In Canada, I switched to "I don't need a bag" or "I have a bag" (notice the construction and syntax of the two sentences are really similar) - it's what everyone else was saying, and I've yet to find a cashier to misunderstand.

I'm tempted to start saying "I already have one" to see if cashiers start misunderstanding. My hunch is that in most abbreviated social situations, people are listening not just for the words, but also for tone, cadence, and general "sound". We don't hear words, we hear sounds which then get parsed into meaning and subsequently action. The actual literal meaning of words is just one form of encoding - often not the primary form, even.

There's no universal "most correct English" - it's all relative.

If you're struggling to be comprehended in spoken English, the best way to avoid the situation is to hold up a bag and jiggle it so that it makes a rustling noise (preferably while smiling and making eye contact).

The cashier will understand completely even if you don't accompany it with the customary "no thanks" or "I've got one/some".

To make it even more clear, in addition to saying No, thanks: visually show that you already have a bag. When I clearly have my backpack already opened in front of me, cashiers often refrain completely from asking whether or not I need a bag. This works even in a country where I don't speak the tiniest bit of the language.

On the other hand, it does not further the goal of improving oral communication skills.

No. I don't need one.
Yes, I want one.