I logged onto an old iPhone with my Apple ID and discovered that it was receiving my text messages

Set up text message forwarding

1. On all your iOS devices, go to Settings > Messages > Send & Receive. On your Mac, open Messages, choose Messages > Preferences, then click iMessage. Make sure that you're signed in to iMessage with the same Apple ID on all your devices.
2. On your iPhone, go to Settings > Messages > Text Message Forwarding.*
3. 
4. Choose which devices can send and receive text messages from your iPhone. If you're not using two-factor authentication for your Apple ID, a verification code appears on each of your other devices: enter that code on your iPhone.

As long as your iPhone is turned on and connected to Wi-Fi or a cellular network, new SMS/MMS texts can be sent and received on the devices that you added.

iCloud now keeps your entire messages history updated and available on all your devices—even when you set up a new device. Learn how to keep all your messages in iCloud.

Don't see the Text Message Forwarding setting on your iPhone? Go to Settings > Messages, turn off iMessage, turn it back on, tap Send & Receive, tap Use Your Apple ID for iMessage, then sign in with the same Apple ID used on your other devices.

How to forward text messages on your iPhone - Apple Support

https://support.apple.com/en-us/HT208386

If you logged out of the old iPhone, it should no longer receive text messages through Apple, aka iMessages.

However, I could imagine, if the old iPhone had the same phone number, it might be possible to receive SMS messages through your cellular carrier, but on second thought, your carrier usually deactivates your old phone's cellular access when you activate a new phone, so you should be good.

Did you log out of Messages/FaceTime on the phone? If you did that, the phone shouldn't receive your messages.