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It really sounds like your SMS app does not have permission to access your contacts. While you say that you've checked that it does, this leads me to two pieces of advice.
First be sure that the correct SMS app has permission to access contacts. Devices usually come with one version of typical apps, SMS, Email, calendar etc. but some devices may come with multiples of these apps and the listing name may be similar between the variants. So just be sure that the app you've chosen as default is the app that you've given permission to.
The second thing that crosses my mind is during updates and some times for seemingly no reason apps act up. The fixes for this can be as simple as clearing the app cache or a little more involved where the app is uninstalled and then installed again. For pre-installed apps one can usually only uninstall the updates and then allow those updates to install again. To do this; Settings>Apps>(SMS app name)>Storage>Clear Cache
Let us know what phone you have?
What and how you've tried to correct this, beyond what you've mentioned?
Are these Google Contacts, phone contacts or contacts pulled from another source?
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