Not receiving text messages

What's the exact model number of the phone, and which carrier are you using? Is it only MMS messages that are having problems (i.e., picture texts or group texts), or even simple texts between two people?
 
Given it's a new phone but old SIM card, are you sure that the SIM is actually working with your current phone? Can you send and receive phone calls like you used to, or is it a matter where phone calling is working but text messaging is not? If you can't do either, perhaps you need to log into your account on your cellular service web site, and check its status. Since your SIM card is how your phone authenticates itself to your cellular service, if your carrier ties your account to a particular device you might need to just update your current status (your new Moto G Stylus). You 'might' need to also update/edit the APN settings in your Moto with that SIM card, typically that info/data is retained within the SIM but since you apparently pulled the card from your previous Moto G7 and put it into your current Moto G Stylus, that might involve editing the APN settings. Your carrier will most likely have the necessary APN data on its site.
 
Welcome to Android Central! I would try downloading a third-party SMS app like Textra, and see if the issue persists.

Turn your phone into Airplane Mode/turn off your network settings and turn them on again, and see what happens.
 

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