Hi Daily_llama, thank you for posting in Apple Support Communities.
Can I suggest your Force Restart your iPad as sometimes that can resolve some legacy issues when updating iPadOS. The following link will explain this more:
https://support.apple.com/en-au/guide/ipad/ipad9955c007/ipados
The same process can also help with your iPhone:
https://support.applehttps://support.apple.com/en-au/HT208386
I have also added the following links to recheck the settings on your iPad:
https://support.apple.com/en-au/guide/ipad/ipad70482978/ipados
And also on your iPhone:
https://support.apple.com/en-au/HT208386
Sometimes turning them off and on can also resolve the issue.
And the following link has more things to try when you cannot send or receive messages:
https://support.apple.com/en-au/HT204065
If the issue persists, please contact Apple Support as follows: