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There is no reason why SMS should be tied to your data package. Turning data roaming off, indeed just turning data off (if you are not currently roaming and merely want to prove this) should still allow you to send and receive SMS messages as these are sent through the voice connection, not the data one - it's exactly how phones have always been able to send a text before anyone ever had data connections.

So the problem is almost certainly something to do with your config or usage, rather than how the SMS systems works. I would ask you to check who you are sending to and from (are you using email addresses rather than phone numbers in messages, either as recipients or your sent from address?), and whether you have Settings > Messages > Send as SMS *ON*

A majority of the time the issue is with a random app or service running in the background. Less often it's because the carrier has to update their software on the device. In both cases, turning off cellular data would solve this issue.

Occasionally there is a problem with the phone itself. Go to Settings>Cellular and note how much cellular data is being used. Then turn off Cellular Data. After two or three hours check if cellular data usage has changed. If it has, restore the iPhone, and if necessary replace it if you have the option.

If there is no change in the cellular data usage I would contact your carrier and let them know.