Then you are postpaid (post = monthly)
For your second concern no need to move elsewhere. Not sure what phone you had before with 7/11 but iPhones need data in order to be able to send/receive iMessage (blue messages). No matter what the provider is, you need data. Disabling iMessage is a pretty easy step you can do with your kid’s phone. Here’s the procedure : https://selfsolve.apple.com/deregister-imessage/
Like I said, not sure you moving to another provider will solve the issue as this is how an iPhone works. Unless they set up a special port for iMessage to go through but not normal data.
Check out Public Mobile. It has MMS included. I am sure @Robert T knows more about iMessage than I do, but I believe it doesn’t use THAT much data and with Public, should you run out, then data add-ons are fairly cheap (and carry over month to month if part of it remains unused).
With PM you pay $15 as well but in my opinion, especially for youngsters, you’d be far better off with that plan.
It doesn’t use a lot of data indeed. But the issue is that once data is activated kids will use it for other reasons leading to iMessage having no more data once again (= parents paying for more and more data). That’s why I’m suggesting to disable it since it doesn’t bring much functionality if you only have 1 device.
You can also go into your iMessage Settings and make. Sure the option to send as sms if iMessage is unavailable. This way the iMessage will convert to regular green text messages when data is stopped.
Keep in mind that if the senders do not have this option activated, the kid without data will still not receive the text messages. Make sure you activate this option on both phones (sender & receiver)